# Charge sensing the parity of an Andreev molecule

David van Driel  Bart Roovers  Francesco Zatelli  Alberto Bordin  Guanzhong
Wang  Nick van Loo  Jan Cornelis Wolff  Grzegorz P. Mazur  QuTech and Kavli
Institute of NanoScience, Delft University of Technology, 2600 GA Delft, The
Netherlands  Sasa Gazibegovic  Ghada Badawy  Erik P. A. M. Bakkers  Department
of Applied Physics, Eindhoven University of Technology, 5600 MB Eindhoven, The
Netherlands  Leo P. Kouwenhoven<EMAIL_ADDRESS>Tom Dvir  QuTech
and Kavli Institute of NanoScience, Delft University of Technology, 2600 GA
Delft, The Netherlands

###### Abstract

The proximity effect of superconductivity on confined states in semiconductors
gives rise to various bound states such as Andreev bound states (ABSs),
Andreev molecules and Majorana zero modes. While such bound states do not
conserve charge, their Fermion parity is a good quantum number. One way to
measure parity is to convert it to charge first, which is then sensed. In this
work, we sense the charge of ABSs and Andreev molecules in an InSb-Al hybrid
nanowire using an integrated quantum dot operated as a charge sensor. We show
how charge sensing measurements can resolve the even and odd states of an
Andreev molecule, without affecting the parity. Such an approach can be
further utilized for parity measurements of Majorana zero modes in Kitaev
chains based on quantum dots.

## Introduction

Majorana zero modes (MZMs) are predicted to appear at the ends of a 1D chain
of spin-polarized, electronic sites with superconducting pairing [1, 2, 3],
which can be implemented using quantum dots (QDs) coupled to superconductors
[4, 5]. Even a minimal, two-site Kitaev chain hosts MZMs in a parameter sweet
spot [6, 7], and was recently realized in a semiconductor-superconductor
hybrid nanowire [8]. MZMs in Kitaev chains are predicted to be robust to local
perturbations and obey non-Abelian statistics, allowing for the demonstration
of Ising anyon fusion rules and braiding [9, 10]. Qubit states can be encoded
in the parity of pairs of MZMs, making parity readout crucial for any quantum
information experiment involving MZMs. Proposed readout techniques include
circuit quantum electrodynamics [11, 12], quantum capacitance [13] and charge
sensing [14]. To read out parity using a charge measurement, it must first be
converted into charge, and then sensed [15]. Although charge sensing has been
applied to semiconductor-superconductor hybrids before, it has never been used
to detect fractional charge differences [16, 17]. This can potentially be
necessary for parity readout in Kitaev chains, given that the charge
difference between the even and odd ground states can range from 0 to $1e$. In
this work, we present charge sensing measurements of a hybrid semiconductor-
superconductor system. First, we measure the charge of an Andreev bound state
(ABS) in its even and odd Fermion parity ground states. Then, we couple a QD
to an ABS to form an Andreev molecule, and infer its ground state parity from
tunnel spectroscopy and charge sensing measurements. As opposed to transport,
charge sensing does not alter the parity itself, highlighting its potential as
a tool for MZM parity readout in Kitaev chains.

## Results

### Device characterization

Figure 1: Device set-up and characterization. a. False color SEM micrograph of
our device. An InSb nanowire (green) is placed on an array of bottom gates
(brown) and contacted by normal Cr/Au leads (yellow). We define a quantum dot
(QD) to the left of the hybrid segment, and operate it as a charge sensor
(CS). Another QD can be formed using the gates to the right of the hybrid
segment. The names of the gate voltages are indicated in the respective gates.
The scale bar is $200\text{\,}\mathrm{nm}$. b. DC equivalent circuit of our
device. Both normal leads of the hybrid segment are connected to off-chip LC
resonators, which are multiplexed. Each LC resonator has a bias tee that can
be used to bias the leads with respect to the grounded Al. c. Amplitude of the
reflected RF signal of the right lead, $V_{\mathrm{RF}}^{\mathrm{R}}$, for
varying voltage on the hybrid plunger gate, $V_{\mathrm{H}}$ and the right
bias, $V^{\mathrm{R}}_{\mathrm{bias}}$, for an external magnetic field
$B=$100\text{\,}\mathrm{m}\mathrm{T}$$, $V_{\mathrm{TR}}=0$ and
$V_{\mathrm{RO}}=$500\text{\,}\mathrm{m}\mathrm{V}$$. Superimposed line: Shift
of the gate voltage corresponding to the Coulomb resonance of the CS, $\Delta
V_{\mathrm{CS}}$, for each $V_{\mathrm{H}}$ setting, extracted from e. The
Zeeman splitting of the first ABS is indicated by
$\Delta\mathrm{E}_{\mathrm{Z}}$. 0, $2e$ and $4e$ indicate additional charge
accumulated on the hybrid. d. Amplitude of the reflected RF signal of the left
lead, $V_{\mathrm{RF}}^{\mathrm{L}}$, for varying voltage on the CS plunger
gate, $V_{\mathrm{CS}}$, at fixed values of $V_{\textrm{H}}$ indicated by the
colored bars in panels c and e. e. $V_{\mathrm{RF}}^{\mathrm{L}}$ for varying
$V_{\mathrm{CS}}$ and $V_{\textrm{H}}$.

Fig. 1a shows an SEM image of the device measured in this work. An InSb
nanowire is placed on a thin layer of gate dielectric, below which are finger
gates. See Refs. [18, 19] for details on device fabrication. The middle
section of the wire, the hybrid segment, is contacted by a grounded Al thin
film and hosts ABSs. We define a QD to the left of the hybrid segment by
setting $V_{\mathrm{LO}}$ and $V_{\mathrm{TL}}$ to create tunnel junctions in
the nanowire. We control the QD’s electrochemical potential using
$V_{\mathrm{CS}}$ and operate it as a single lead QD charge sensor (CS) [20].
The tunnel gate between the CS and hybrid segment is kept at a negative
voltage $V_{\mathrm{TL}}=$-100\text{\,}\mathrm{m}\mathrm{V}$$ to fully quench
transport and ensure their coupling is only capacitive. The nanowire section
to the right of the hybrid segment can be either a tunnel barrier or a QD,
depending on the tunnel gate voltages $V_{\mathrm{RO}}$ and $V_{\mathrm{TR}}$.

The nanowire is contacted by two normal Cr/Au leads that can be used for DC
transport and radio frequency (RF) reflectometry measurements. Fig. 1b shows
the DC equivalent circuit of the device and its connections. Both normal leads
are connected to LC resonators with bias tees, allowing us to independently
voltage bias them with respect to the grounded Al. The LC resonators are off-
chip and multiplexed, see Ref. [21] for details. Further details on the
reflectometry circuit can be found in Ref. [22]; for resonator
characterization, see Sec. I of the Supplemental Material [23]. Experiments
are performed in a dilution refrigerator at a base temperature of
$30\text{\,}\mathrm{mK}$.

### ABS even ground state charge

The hybrid segment hosts ABSs that can have an odd, doublet ground state
($\\{\ket{\downarrow},\ket{\uparrow}\\}$) or an even ground state. We consider
an ABS in the atomic limit, where the even ground state is a singlet:
$\ket{S}=u\ket{0}-v\ket{2}$, with 0 and 2 denoting the occupation of a single
orbital [24, 25]. The singlet state changes between being mostly unoccupied
(0-like) or doubly occupied (2-like) when the electrochemical potential of the
ABS changes. Note that this is a gradual change of ground state, as opposed to
changing a QD ground state by occupying additional electrons. When the
semiconductor-superconductor coupling is stronger than the charging energy,
the ABS has an even ground state with a doublet excited state [26].

To characterize the hybrid segment, we set
$V_{\mathrm{RO}}=$500\text{\,}\mathrm{m}\mathrm{V}$$ to accumulate electrons,
and create a tunnel barrier with $V_{\mathrm{TR}}$ to perform tunneling
spectroscopy from the right lead. Fig. 1c shows amplitude of the reflected RF
signal of the right lead, $V_{\mathrm{RF}}^{\mathrm{R}}$, for varying hybrid
plunger gate voltage, $V_{\mathrm{H}}$, and right voltage bias,
$V^{\mathrm{R}}_{\mathrm{bias}}$. A dip in $V_{\mathrm{RF}}^{\mathrm{R}}$ can
be related to a peak in differential conductance [22]. The hybrid segment has
a hard-gapped density of states with two well-separated ABSs of which the
energy can be controlled with $V_{\textrm{H}}$. Each ABS excitation is split
into two resonances, because we apply $100\text{\,}\mathrm{m}\mathrm{T}$ along
the nanowire axis. From the Zeeman splitting $\Delta
E_{\mathrm{Z}}=$80\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$, we obtain an
effective $g$-factor of $g=13.8$ for the ABS. The ABS excitations do not cross
zero bias, signalling an even ground state for the entire $V_{\textrm{H}}$
range.

Fig. 1d shows amplitude of the reflected RF signal of the left lead,
$V_{\mathrm{RF}}^{\mathrm{L}}$, for varying plunger gate voltage of the CS,
$V_{\mathrm{CS}}$, taken at three different values of $V_{\mathrm{H}}$. A
Coulomb resonance of the CS is seen as a dip in $V_{\mathrm{RF}}^{\mathrm{L}}$
whenever a CS level is aligned with the Fermi level of the left lead [27]. We
see that the $V_{\mathrm{CS}}$ value corresponding to the
$V_{\mathrm{RF}}^{\mathrm{L}}$ minimum changes due to $V_{\textrm{H}}$, which
has two possible causes. First, $V_{\textrm{H}}$ directly gates the CS due to
cross-capacitance. Second, $V_{\textrm{H}}$ can change the occupation of the
hybrid segment and the resulting charge is sensed by the CS. To extract the
charge sensing signal only, we define virtual gates that are linear
combinations of the physical gate voltages, to compensate for cross-
capacitances between gates (for details see Sec. II of the Supplemental
Material [23]).

In Fig. 1e, we show one particular CS Coulomb resonance for varying
$V_{\mathrm{CS}}$ and $V_{\mathrm{H}}$. We have subtracted a global slope from
the data, which we attribute to imperfect virtual gate settings. The global
slope was chosen such that the CS resonance is roughly constant in
$V_{\textrm{H}}$ up to $300\text{\,}\mathrm{mV}$, for which there are no sub-
gap states (for details and raw data, see Sec. II of the Supplemental Material
[23].

The superimposed black line of Fig. 1c shows the shift of the CS Coulomb
resonance, $\Delta V_{\mathrm{CS}}$, as found from Fig. 1e. We see that
$\Delta V_{\mathrm{CS}}$ depends on $V_{\textrm{H}}$ and changes most strongly
when there is an ABS at sub-gap energies. The absence of sharp jumps in
$\Delta V_{\mathrm{CS}}$ shows that there are no abrupt changes of charge,
which is consistent with an even ground state for the entire $V_{\textrm{H}}$
range as found from Fig. 1c. We interpret the change in $\Delta
V_{\mathrm{CS}}$ at $V_{\textrm{H}}=$310\text{\,}\mathrm{mV}$$ and
$V_{\textrm{H}}=$360\text{\,}\mathrm{mV}$$ as the CS sensing the charge of the
ABSs changing continuously from 0 to $2e$, where $e$ is the charge of the
electron.

Gradual change of charge without change of parity, as seen here, has been
observed before in normal double quantum dots [28, 29]. In our case, however,
the ABS exchanges charge with a large Al reservoir and becomes a coherent
superposition of 0 and $2e$.

### Single ABS parity readout

Figure 2: Charge sensing the even and odd ground states of an ABS. a.-c.
Amplitude of the reflected RF signal of the right lead,
$V_{\mathrm{RF}}^{\mathrm{R}}$, for varying hybrid gate voltage
$V_{\mathrm{H}}$ and right bias $V^{\mathrm{R}}_{\mathrm{bias}}$, taken at
$B=$0\text{\,}\mathrm{m}\mathrm{T}$$ (a.),
$B=$100\text{\,}\mathrm{m}\mathrm{T}$$ (b.) and
$B=$250\text{\,}\mathrm{m}\mathrm{T}$$ (c.) for
$V_{\mathrm{TR}}=$0\text{\,}\mathrm{m}\mathrm{V}$$,
$V_{\mathrm{TL}}=$-100\text{\,}\mathrm{m}\mathrm{V}$$,
$V_{\mathrm{LO}}=$170\text{\,}\mathrm{m}\mathrm{V}$$ and
$V_{\mathrm{RO}}=$500\text{\,}\mathrm{m}\mathrm{V}$$. d. Shift of the CS
Coulomb resonance, $\Delta V_{\mathrm{CS}}$, at different $B$ values for
$V_{\mathrm{TL}}=$-140\text{\,}\mathrm{m}\mathrm{V}$$ and
$V_{\mathrm{LO}}=$172\text{\,}\mathrm{m}\mathrm{V}$$. The horizontal lines are
offset by $75\text{\,}\mathrm{\SIUnitSymbolMicro V}$, which corresponds to a
charge difference of approximately $1e$. The arrows indicate the even-odd
transition at $B=$0\text{\,}\mathrm{m}\mathrm{T}$$ and
$B=$300\text{\,}\mathrm{m}\mathrm{T}$$ respectively. inset. The occupation
expectation value of the ABS ground state, $\langle n\rangle$, for varying
chemical potential, $\mu$, at different $B$ values, calculated in the atomic
limit. The arrows indicate the even-odd transition.

To use our charge sensor for parity readout, we focus on an ABS with a ground
state that can be changed from even to odd parity using $V_{\textrm{H}}$. Fig.
2a shows spectroscopy of such an ABS in a different $V_{\textrm{H}}$ range
than Fig. 1, for $B=$0\text{\,}\mathrm{m}\mathrm{T}$$. We see a sub-gap state
that crosses zero energy twice for changing $V_{\textrm{H}}$, give rising to a
characteristic ABS eye-shape [30, 31]. The ABS ground state between the zero-
energy crossings is a spin doublet $\ket{D}=\ket{\downarrow},\ket{\uparrow}$
[26]. Having a doublet ground state at $B=$0\text{\,}\mathrm{m}\mathrm{T}$$
signals that the ABS has a charging energy that is non-negligible. While this
is uncommon among most ABSs in the hybrid segment, we utilize it to measure
varying parities (for measurements in a larger $V_{\textrm{H}}$ range, see
Sec. III of the Supplemental Material [23]). Fig. 2b and c show spectroscopy
of the ABS at $B=$100\text{\,}\mathrm{m}\mathrm{T}$$ and
$B=$250\text{\,}\mathrm{m}\mathrm{T}$$, respectively. The magnetic field
Zeeman splits the doublet states, making the odd state lower in energy than
the even state over a larger $V_{\textrm{H}}$ range.

In Fig. 2d we show the processed shift of the CS Coulomb resonance, $\Delta
V_{\mathrm{CS}}$, for varying $V_{\textrm{H}}$ at different $B$ values. To
obtain $\Delta V_{\mathrm{CS}}$ for each $B$ value, we first set the CS on
Coulomb resonance using $V_{\mathrm{CS}}$ and measure
$V_{\mathrm{RF}}^{\mathrm{L}}$ for varying $V_{\textrm{H}}$. We then convert
the measured $V_{\mathrm{RF}}^{\mathrm{L}}$ to the corresponding shift of the
CS Coulomb resonance, resulting in $\Delta V_{\mathrm{CS}}$. For details, see
Sec. IV of the Supplemental Material [23].

For each $B$ value, we identify three distinct $V_{\mathrm{H}}$ regions where
the $\Delta V_{\mathrm{CS}}$ response is roughly flat. The middle region of
these around $V_{\mathrm{H}}\approx$260\text{\,}\mathrm{m}\mathrm{V}$$
occupies a larger $V_{\textrm{H}}$ range for increasing $B$. This is
consistent with $\ket{\downarrow}$ being the ground state for a larger
$V_{\textrm{H}}$ range, as inferred from Fig. 2a-c (note that the
$V_{\textrm{H}}$ values for which the ABS has an odd ground state differ
slightly between panels a-c. and d. due to different $V_{\mathrm{LO}}$ and
$V_{\mathrm{TL}}$ gate voltages, which were chosen to optimize the sensitivity
of the charge sensor). We interpret the three distinct $\Delta
V_{\mathrm{CS}}$ values for increasing $V_{\textrm{H}}$ as corresponding to
the 0-like even, singly occupied odd, and 2-like even states of the ABS.

We observe finite curvature in $\Delta V_{\mathrm{CS}}$ for the even ground
state, which we attribute to mixing of the 0 and 2 occupations, similar to
Fig. 1e. The curvature is most visible close to the even-odd transition, and
becomes less apparent with increasing $B$. At
$B=$0\text{\,}\mathrm{m}\mathrm{T}$$, the even-odd transition (indicated by
the blue arrow) occurs when the ABS is near its energy minimum, where the
average ABS charge is $1e$ for both the even and odd states [32, 33]. The even
state charge gradually changes from 0 to almost $1e$ in the $V_{\textrm{H}}$
range before the transition. At $B=$300\text{\,}\mathrm{m}\mathrm{T}$$, the
even-odd transition occurs when the ABS is no longer close to its energy
minimum. The difference in charge between the even and odd states is then
greater, resulting in a sharper change of $\Delta V_{\mathrm{CS}}$, as
indicated by the pink arrow.

If the curvature of $\Delta V_{\mathrm{CS}}$ is exclusive to the even ground
state, we can use it to infer the parity of the ABS using charge sensing only.
To illustrate this point, we calculate and show the charge of an ABS in the
inset of Fig. 2d. Here, the average occupation, $\langle n\rangle$, is shown
for varying chemical potential, $\mu$, at two different values of $B$ (for
more details see Sec. V of the Supplemental Material [23]). We conclude that
at low $B$, curvature in $\langle n\rangle$ is a characteristic sign of an ABS
in a singlet ground state, provided the parent gap is much larger than the
charging energy.

So far, we have established that charge sensing measurements can resolve the
charge differences between the even and odd ground states of a single ABS. In
a two-site Kitaev chain, however, it is the combined parity of two hybridized
QDs that has to be detected. We create a proxy system by hybridizing a QD with
an ABS and charge sense it to give a minimal demonstration of parity readout
for Kitaev chains. Parity readout of Majorana zero modes requires extra steps,
which we detail in the discussion. We note that an alternating ABS-QD array
can also constitute a Kitaev chain [34, 35].

### Andreev molecule spectroscopy

Figure 3: Spectroscopy of a hybridized ABS-QD system. a. Amplitude of the
reflected RF signal of the right lead, $V_{\mathrm{RF}}^{\mathrm{R}}$ for
varying QD plunger gate voltage, $V_{\mathrm{QD}}$, and $V_{\textrm{H}}$ taken
at $V^{\mathrm{R}}_{\mathrm{bias}}=$0\text{\,}\mathrm{V}$$,
$B=$200\text{\,}\mathrm{m}\mathrm{T}$$,
$V_{\mathrm{TR}}=$215\text{\,}\mathrm{m}\mathrm{V}$$ and
$V_{\mathrm{RO}}=$340\text{\,}\mathrm{m}\mathrm{V}$$. b. Parity of the lowest
energy eigenstate of a coupled ABS-QD system for varying QD and ABS chemical
potential $\tilde{\mu}_{\mathrm{QD}}$ and $\tilde{\mu}_{\mathrm{ABS}}$,
computed in the atomic limit. We designate the ground state parity as even if
$E_{\mathrm{Odd}}>E_{\mathrm{Even}}$ and odd if
$E_{\mathrm{Odd}}<E_{\mathrm{Even}}$. The superimposed text indicates the
ground states in the
$\ket{N,M}=\ket{N}_{\mathrm{QD}}\otimes\ket{M}_{\mathrm{ABS}}$ basis of the
ABS-QD system. We denote the ABS singlet state by $\ket{S}$, and highlight its
dominant occupation for the $\ket{\downarrow,S}$ state. c., d.
$V_{\mathrm{RF}}^{\mathrm{R}}$ for varying $V_{\mathrm{QD}}$ and
$V^{\mathrm{R}}_{\mathrm{bias}}$, taken at two different values of
$V_{\textrm{H}}$ indicated by the blue and green horizontal bars in panel a.
e., f. $V_{\mathrm{RF}}^{\mathrm{R}}$ for varying $V_{\textrm{H}}$ and
$V^{\mathrm{R}}_{\mathrm{bias}}$, taken at two different values of
$V_{\mathrm{QD}}$ indicated by the red and yellow vertical bars in panel a.

To create the ABS-QD system, we define a QD to the right of the hybrid segment
by creating a tunnel barrier using $V_{\mathrm{RO}}$ (see Fig. 1a). We then
lower the tunnel barrier between the QD and the hybrid segment by increasing
$V_{\mathrm{TR}}$.

Fig. 3a shows a charge stability diagram measured in
$V_{\mathrm{RF}}^{\mathrm{R}}$ for varying $V_{\textrm{H}}$ and QD plunger
gate, $V_{\mathrm{QD}}$. We see avoided crossings that indicate hybridization
of the QD and ABS. The QD resonance can be seen when the ABS is off-resonance
(blue bar), which we attribute to local Andreev reflection on the QD and the
usage of RF reflectometry instead of differential conductance (for details see
Sec. VI of the Supplemental Material [23]). Similarly, the ABS can be observed
when the QD is off-resonance (e.g., red bar). Because of the strong
hybridization between the QD and ABS, the latter can always be seen in
spectroscopy upon probing the QD.

Fig. 3b shows the ground state parity of a QD coupled to an ABS, computed in
the atomic limit. States are indicated in the ABS-QD number basis, with
$\ket{S}$ denoting the ABS singlet (for details on the model, see Sec. V of
the Supplemental Material [23]). We emphasize that the singlet is dominantly
0-like or 2-like at $-\tilde{\mu}_{\mathrm{ABS}}=\pm 1\ U_{\mathrm{QD}}$. We
see that Fig. 3b qualitatively reproduces most of the features of Fig. 3a.

Fig. 3c and d show spectroscopy of the QD for varying $V_{\mathrm{QD}}$ and
$V^{\mathrm{R}}_{\mathrm{bias}}$ at fixed values of $V_{\textrm{H}}$ indicated
by the horizontal lines in Fig. 3a. When the ABS in the hybrid segment is not
at zero-energy (blue bar, panel c), we see multiple sub-gap states, where the
lowest one forms a typical eye-shape. This indicates the formation of Yu-
Shiba-Rusinov (YSR) states on the QD, due to its strong coupling to the hybrid
segment [36, 37, 38, 39, 40]. Fig. 3d (green bar) shows spectroscopy of the QD
when the ABS excitation is at zero energy, i.e., on-resonance. Here we see
that the ABS and QD states form sub-gap bonding and anti-bonding states when
they are on resonance, indicated by the blue arrows. Two hybridized sub-gap
states are often referred to as an “Andreev molecule”, which usually
designates two coupled YSR states [41, 42, 43, 44, 45, 46, 47] or phase-
tunable ABSs in Josephson junctions [48, 49, 50]. Because our ABS-QD system
shows hybridization of an ABS in a hybrid with a YSR state in a QD, we
categorize it as an Andreev molecule.

Fig. 3e and f show spectroscopy of the QD for varying $V_{\textrm{H}}$ at
fixed values of $V_{\mathrm{QD}}$ indicated by the vertical lines in Fig. 3a.
When the QD is off-resonance (red bar, panel e) the lowest sub-gap state
crosses zero energy twice, and the excited states are close in energy to the
superconducting gap. Fig. 3f (yellow bar) shows spectroscopy when the QD
excitation is at zero-energy. When the ABS is on resonance, it splits the
zero-bias peak of the QD, indicated by the white arrows. This effect is known
to occur for QDs strongly coupled to ABSs [51, 52, 53].

### Andreev molecule parity readout

Figure 4: Charge sensing measurements of the ABS-QD system. All data was taken
for $V_{\mathrm{TR}}=$255\text{\,}\mathrm{m}\mathrm{V}$$,
$V_{\mathrm{RO}}=$230\text{\,}\mathrm{m}\mathrm{V}$$ and
$B=$250\text{\,}\mathrm{m}\mathrm{T}$$. a. Phase of the reflected RF signal of
the left lead, $\phi^{\mathrm{L}}_{\mathrm{RF}}$, for varying
$V_{\mathrm{QD}}$ and $V_{\textrm{H}}$. b. Zoom in of a. for the range
indicated by the dashed rectangle. The superimposed text indicates the ground
state of the ABS-QD system in the same basis as in Fig. 3b. c. The numerical
derivative, $d\phi^{\mathrm{L}}_{\mathrm{RF}}/dV_{\textrm{H}}$, of b. The
superimposed text indicates the parity of the states in panel b, where ”O” and
”E” indicate an odd and even ground state, respectively. The two crosses
indicate negative peaks of $d\phi^{\mathrm{L}}_{\mathrm{RF}}/dV_{\textrm{H}}$
found from peak-finding for the cut along the orange diagonal line shown in
panel b. d. $\phi^{\mathrm{L}}_{\mathrm{RF}}$ for a cut along the orange
diagonal line shown in panel b. Here, $V_{\mathrm{QD}}$ and $V_{\textrm{H}}$
are changed in parallel, indicated by $\delta$. The vertical black lines
correspond to the $V_{\mathrm{QD}}$ and $V_{\textrm{H}}$ values of the black
crosses in panel c, from which we infer a parity change. The red and blue
shading indicate even and odd ground states, respectively.

Next, we sense the charge of the ABS within the ABS-QD system. Fig. 4a shows
the phase of the reflected RF signal of the left lead,
$\phi^{\mathrm{L}}_{\mathrm{RF}}$, for varying $V_{\mathrm{QD}}$ and
$V_{\textrm{H}}$, at different gate and field settings from Fig. 3. We set the
CS on Coulomb resonance using $V_{\mathrm{CS}}$, at the $V_{\mathrm{QD}}$ and
$V_{\textrm{H}}$ settings that correspond to the bottom left corner of Fig.
4a. Globally, we observe three horizontal regions where
$\phi^{\mathrm{L}}_{\mathrm{RF}}$ depends almost only on $V_{\textrm{H}}$,
which we interpret as the 0-like even, singly occupied $\ket{\downarrow}$ and
2-like even states of the ABS. In addition, there are 4 regions where
$V_{\mathrm{RF}}^{\mathrm{L}}$ depends visibly on $V_{\mathrm{QD}}$, which we
attribute to the QD hybridizing with the ABS. These regions are highlighted
using dashed rectangles.

In Fig. 4b we show a zoom-in of panel a corresponding to the black rectangle.
Here, the CS was gated to be on the steepest slope of the Coulomb peak using
$V_{\mathrm{CS}}$ at the start of the measurement. We see an off-diagonal
avoided crossing, which signals an interdot transition [54]. Because this is
the first interdot transition of this QD orbital and ABS, we label the bottom
left corner of Fig. 4b with $\ket{0,S}$, where the QD orbital is unoccupied
and the ABS is in the 0-like singlet state. The other states are labeled by
counting the added electrons. Although the QD is proximitized by coupling to
the hybrid, as seen in Fig. 3c, we retain the number basis for clarity.

Fig. 4c shows the numerical derivative of panel b,
$d\phi^{\mathrm{L}}_{\mathrm{RF}}/dV_{\textrm{H}}$, after processing with a
Savgol filter. The dips in $d\phi^{\mathrm{L}}_{\mathrm{RF}}/dV_{\textrm{H}}$
form two blue hyperbolas that indicate where the even and odd ground states
are degenerate [29]. Based on the states inferred from panel b, we label the
region between the hyperbolas as odd (”O”) and the two areas outside as even
(”E”).

Fig. 4d shows a linecut of $\phi^{\mathrm{L}}_{\mathrm{RF}}$ taken along the
orange line in panel b, where $V_{\mathrm{QD}}$ and $V_{\textrm{H}}$ are
changed in parallel, indicated by $\delta$. The two vertical, dashed black
lines correspond to the gate values of the superimposed crosses in panel c,
which we use to divide the range of $\delta$ into even and odd ground states.
We note that there are three regions where $\phi^{\mathrm{L}}_{\mathrm{RF}}$
is roughly constant, and that these coincide with even and odd parity sectors.
From this we conclude that we can read out the parity of the ABS-QD system by
measuring its charge.

Extending parity readout using charge sensing to a two-site Kitaev chain
requires an additional step however. In the ”Poor Man’s Majorana” sweet spot,
the degenerate even and odd parity ground states have the same charge [7].
Detuning one QD creates a charge difference between the even and odd ground
states on the other QD, while the states remain degenerate [55]. A local
charge sensing measurement on the non-detuned QD can then tell the parity
based on charge. This protocol of detuning and measuring charge has to be
performed within the quasi-particle poisoning time, otherwise the parity flips
during the measurement. We note that our RF integration time of
$\tau=$9.3\text{\,}\mathrm{ms}$$ is on the order of quasiparticle poisoning
times typically found in hybrid systems, although estimates vary strongly
depending on device design [56, 57]. This results in the CS predominantly
sensing the average occupation of our system. We show SNR values for different
integration times in Sec. VII of the Supplemental Material [23].

## Conclusion

In conclusion, we have operated a QD as a charge sensor and measured the
charge of ABSs in a hybrid semiconductor-superconductor nanowire. We have
found that the charge of an ABS can change by $2e$ while remaining in the even
ground state. The charge difference between the even and odd states can be
less than $1e$ due to the ABS exchanging charge with the superconductor. We
have coupled a QD to an ABS to form a hybridized state, which we categorize as
an Andreev molecule. Using the charge sensor, we can infer the parity of the
ABS-QD system when the even and odd ground states have different charges. For
a two-site Kitaev chain, both parities have the same charge in the sweet spot,
and require an additional step for parity-to-charge conversion. We demonstrate
that charge sensing can be used for parity readout in hybrid systems and is
promising for usage in Kitaev chains.

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## Acknowledgements

This work has been supported by the Dutch Organization for Scientific Research
(NWO) and Microsoft Corporation Station Q. We wish to acknowledge useful
discussions with Chun-Xiao Liu and Anasua Chatterjee.

## Author contributions

DVD, GW, AB, NvL, FZ, GM fabricated the devices. DVD and BR performed the
electrical measurements. DVD, TD and BR designed the experiment and analyzed
the data. DVD, TD and LPK prepared the manuscript with input from all authors.
TD and LPK supervised the project.

## Data availability

All raw data in the publication and the analysis code used to generate figures
are available at https://doi.org/10.5281/zenodo.10067038.

## Supplemental Material

###  I Resonator Characterization

Figure S1: Resonator characterization. Fits of off-chip resonators performed
according to the model described in Ref. [58]. a. Amplitude of the reflected
RF signal of the left lead, $V_{\mathrm{RF}}^{\mathrm{L}}$, versus the
frequency of the RF output signal, $f$. b. Amplitude of the reflected RF
signal of the right lead, $V_{\mathrm{RF}}^{\mathrm{R}}$, versus the frequency
of the RF output signal, $f$.

Both normal contacts of the device presented in this work are bonded to off-
chip LC resonators. Fig. S1a,b show the amplitude of the reflected RF signal
of the left and right lead $V^{\mathrm{L/R}}_{\mathrm{RF}}$ for varying
frequency, $f$, at $B=$0\text{\,}\mathrm{m}\mathrm{T}$$. We fit the resonator
responses using a model for asymmetric resonances by Khalil et al. [58]. From
the fit, we obtain the internal and external Q-factors listed in the figure.
For all other measurements in this work, we fix the frequencies at
$f_{\mathrm{L}}=$364\text{\,}\mathrm{MHz}$$ and
$f_{\mathrm{R}}=$264\text{\,}\mathrm{MHz}$$. All RF measurements were
performed using a Zurich Instruments UHFLI.

###  II Virtual Gates and Fig. 1 Data Processing

Figure S2: Virtual gates. a. Device sketch that highlights the physical gates
used to define the new virtual gates b. The cross-capacitance matrix for the
gates highlighted in a. Off-diagonal numbers indicate a finite cross-coupling
between physical gates, which is corrected for using the virtual gates. This
setting is used for Fig. 1 and 2 . c. Same as panel b, but corresponding to
the data from figure 4 of the main text.

Fig. S2 shows the gates and cross-capacitance matrix $\hat{C}$ used for the
virtual gates in this work. The matrix $\hat{C}$ allows us to estimate the
actual electrochemical potential change due to cross-capacitance, using:
$\vec{V}^{\prime}=\hat{C}\vec{V}$. Here, the vector $\vec{V}$ contains all the
physical gate voltages and $\vec{V}^{\prime}$ is the resulting voltage on
parts of the device. This allows us to change gates in a way that, for
example, affects only the QD and not the CS electrochemical potential. For
more details on virtual gates, see Refs. [59, 60].

Our virtual gate implementation was imperfect, leading to a residual cross-
capacitance between $V_{\textrm{H}}$ and $V_{\mathrm{CS}}$. Fig. S3a shows
that the CS resonance is also gated by $V_{\textrm{H}}$. We attribute the
residual cross-coupling to the non-linearity of the device and an inaccurate
choice of $\hat{C}$. We correct for this in post-processing by subtracting a
global slope, until the CS resonance does not depend on $V_{\textrm{H}}$ up to
$V_{\textrm{H}}=$300\text{\,}\mathrm{mV}$$. This result is presented in Fig.
1e and in Fig. S3b. In Fig. S3d we show the minima of the Coulomb dip for each
$V_{\textrm{H}}$ value, extracted from panels a., b. The processed data is
seen to be devoid of a global slope. The line in Fig. S2d is superimposed in
Fig. 1c.

Figure S3: Data processing of Fig. 1. a. Amplitude of the reflected RF signal
of the left lead, $V_{\mathrm{RF}}^{\mathrm{L}}$, for varying
$V_{\mathrm{CS}}$ and $V_{\textrm{H}}$. White markers indicate the minimum of
the Coulomb resonance for each $V_{\textrm{H}}$ value. b. Same as a., but
after removing a global slope between $V_{\mathrm{CS}}$ and $V_{\textrm{H}}$.
c.,d. Shift of the CS Coulomb resonance, $\Delta V_{\mathrm{CS}}$, for each
$V_{\mathrm{H}}$ setting, extracted from a. and b. respectively. These
correspond to the white markers in panels a and b. The line of panel d is
superimposed on Fig. 1c.

###  III Extended Range Hybrid Spectrum and Charge Sensing

Fig. S4a shows spectroscopy of the hybrid for a larger range of
$V_{\textrm{H}}$ than shown in Fig. 1c. The ABSs shown in Fig. 1 are indicated
by the blue marks. The ABS of Fig. 2 is indicated by the white marks. Panel b
shows the corresponding charge sensor measurement.

Figure S4: Extended $V_{\textrm{H}}$ range measurement of Fig. 1. a. Amplitude
of the reflected RF signal of the right lead, $V_{\mathrm{RF}}^{\mathrm{R}}$
for varying hybrid gate voltage $V_{\mathrm{H}}$ and right bias
$V^{\mathrm{R}}_{\mathrm{bias}}$ at $B=$100\text{\,}\mathrm{m}\mathrm{T}$$,
$V_{\mathrm{TR}}=0$ and $V_{\mathrm{RO}}=$500\text{\,}\mathrm{m}\mathrm{V}$$.
Superimposed line. Shift of the CS Coulomb resonance for each $V_{\mathrm{H}}$
setting, extracted from b. A global slope is subtracted for clarity. b.
Amplitude of the reflected RF signal of the left lead,
$V_{\mathrm{RF}}^{\mathrm{L}}$, for varying $V_{\mathrm{CS}}$ and
$V_{\textrm{H}}$. White dots indicate the CS resonance found from peak-
finding. The ABSs shown in Fig. 1 are indicated by the blue marks. The ABS of
Fig. 2 is indicated by the white marks. Panel b shows the corresponding charge
sensor measurement.

###  IV Fig. 2 Charge Sensor Data Processing

Figure S5: Data processing of Fig. 2. a-g. Amplitude of the reflected RF
signal of the left lead, $V_{\mathrm{RF}}^{\mathrm{L}}$, for varying
$V_{\mathrm{CS}}$ at $V_{\textrm{H}}=$234\text{\,}\mathrm{m}\mathrm{V}$$ for
values of $B$ indicated in the title. These measurements are used to map a
measured $V_{\mathrm{RF}}^{\mathrm{L}}$ value to the closest $V_{\mathrm{CS}}$
value. Orange markers. The $V_{\mathrm{RF}}^{\mathrm{L}}$ values from the data
shown in panels h-n, mapped to $V_{\mathrm{CS}}$. For each $V_{\textrm{H}}$
setpoint, the measured $V_{\mathrm{RF}}^{\mathrm{L}}$ is compared to the full
$V_{\mathrm{RF}}^{\mathrm{L}}$ v.s. $V_{\mathrm{CS}}$ curve of panels a-g.
Each measured $V_{\mathrm{RF}}^{\mathrm{L}}$ value from panels h-n is then
mapped to a $V_{\mathrm{CS}}$ value from panels a-g. h-n.
$V_{\mathrm{RF}}^{\mathrm{L}}$ for varying $V_{\textrm{H}}$ at fixed
$V_{\mathrm{CS}}$ for values of $B$ indicated in the title. At each value of
$V_{\textrm{H}}$, the measured $V_{\mathrm{RF}}^{\mathrm{L}}$ is compared to
the corresponding measurement of panels a-g. for the same $B$. The closest
matching value of $V_{\mathrm{CS}}$ is then chosen, resulting in a mapping of
$V_{\textrm{H}}$ to $V_{\mathrm{CS}}$, which is plotted in orange in panels
a-g. o-u. Shift of the gate voltage corresponding to the Coulomb resonance of
the CS, $\Delta V_{\mathrm{CS}}$, for varying $V_{\textrm{H}}$. These
correspond to the orange markers of panels a-g. after subtracting a global
slope.

A change of the ABS charge results in a linear shift of the CS electrochemical
potential, proportional to their mutual capacitance. The resulting change in
reflected signal $V_{\mathrm{RF}}^{\mathrm{L}}$ is not linear due to the shape
of the Coulomb dip. To compensate for this, we first perform a
characterization measurement of $V_{\mathrm{RF}}^{\mathrm{L}}$ for varying
$V_{\mathrm{CS}}$ for each $B$ value as shown in Fig. S5a-g. We then fix the
$V_{\mathrm{CS}}$ gate value such that the CS is on Coulomb resonance and
measure $V_{\mathrm{RF}}^{\mathrm{L}}$ while varying $V_{\mathrm{H}}$, which
is shown in Fig. S5h-n. For each measured $V_{\mathrm{RF}}^{\mathrm{L}}$ value
per $V_{\textrm{H}}$, we find the closest $V_{\mathrm{RF}}^{\mathrm{L}}$ value
in the characterization measurement. We then map the measured
$V_{\mathrm{RF}}^{\mathrm{L}}$ value to $V_{\mathrm{CS}}$, which is indicated
by the orange markers in Fig. S5a-g. Finally, we subtract a global slope from
the resulting $V_{\mathrm{CS}}$ value that we attribute to remaining cross-
capacitance to $V_{\textrm{H}}$ and show the resulting $\Delta
V_{\mathrm{CS}}$ in Fig. S5o-u. We note that the vertical difference between
the three plateaus is now roughly equal, as compared to Fig. S5h-n. Therefore,
we interpret the processed data as being proportional to the ABS charge.

###  V Andreev Bound States In the Atomic Limit

The inset of Fig. 2d shows the occupation of an ABS modeled in the atomic
limit. In the many-body basis of a single orbital
$\\{\ket{0},\ket{2},\ket{\downarrow},\ket{\uparrow}\\}$ the Hamiltonian is:

$H_{\mathrm{ABS}}=\begin{bmatrix}0&-\Gamma&0&0\\\ -\Gamma&2\mu+U&0&0\\\
0&0&\mu+E_{\mathrm{Z}}&0\\\ 0&0&0&\mu-E_{\mathrm{Z}}\\\ \end{bmatrix}$

Where $\mu$ is the electrochemical potential of the uncoupled orbital,
$\Gamma$ is the coupling to the superconductor, $U$ is the charging energy and
$E_{\mathrm{Z}}$ is the Zeeman energy. We diagonalize the Hamiltonian, obtain
the eigenvectors and calculate the ABS ground state occupation $\langle
n\rangle$. For Fig. 2d, we use $\Gamma=2.45$, $U=5$ and
$E_{\mathrm{Z}}=0,0.25,0.6$. For more details on the model, see Refs. [61,
62].

For Fig. 3, we combine an atomic-limit ABS with a QD:

$H=H_{\mathrm{ABS}}+H_{\mathrm{QD}}+H_{t}$ (1)

Where for the QD we have:

$H_{\mathrm{QD}}=(\mu_{\mathrm{QD}}-E_{\mathrm{Z}})c_{\downarrow}^{\dagger}c_{\downarrow}+U_{\mathrm{QD}}c_{\uparrow}^{\dagger}c_{\uparrow}c_{\downarrow}^{\dagger}c_{\downarrow}.$
(2)

$H_{t}$ describes the tunnel coupling between the ABS and the QD and is given
by:

$H_{t}=t(c_{\downarrow}^{\dagger}d_{\downarrow}+c_{\uparrow}^{\dagger}d_{\uparrow}+h.c.)+t_{\mathrm{so}}(-c_{\downarrow}^{\dagger}d_{\uparrow}+c_{\uparrow}^{\dagger}d_{\downarrow}+h.c.),$
(3)

Where $t$ is the spin-conserving tunnelling amplitude, and $t_{\mathrm{so}}$
is the spin-flipping tunneling amplitude resulting from spin-orbit
interaction.

The basis of Eq. 1 can be split into even and odd parity subspaces, where we
consider the following states in the
$\ket{N,M}=\ket{N}_{\mathrm{QD}}\otimes\ket{M}_{\mathrm{ABS}}$ basis:

$\displaystyle\\{\ket{0,S},\ket{2,S},\ket{\downarrow,\downarrow}\\}$
$\displaystyle\mathrm{even}$
$\displaystyle\\{\ket{0,\downarrow},\ket{\downarrow,S},\ket{2,\downarrow}\\}$
$\displaystyle\mathrm{odd}$

Here $\ket{S}=u\ket{0}-v\ket{2}$ is the ABS singlet. Using Eq. 3, we can
calculate the effective coupling terms between the basis states of the system.
We only consider transitions with a fixed global parity, i.e. only transitions
between two odd occupation states or transitions between two even occupation
states. The effective coupling between the different even occupation states of
the system is given by:

$\braket{2,S}{H_{t}}{0,S}=0$ (4)
$\braket{\downarrow,\downarrow}{H_{t}}{0,S}=vt_{\mathrm{so}}$ (5)
$\braket{2,S}{H_{t}}{\downarrow,\downarrow}=-ut_{\mathrm{so}}$ (6)

For the odd occupation states we have:

$\braket{0,\downarrow}{H_{t}}{\downarrow,S}=ut$ (7)
$\braket{0,\downarrow}{H_{t}}{2,\downarrow}=0$ (8)
$\braket{2,\downarrow}{H_{t}}{\downarrow,S}=vt$ (9)

We can now write down Equation 1 in the even and odd parity subspace matrix
representation:

$H_{even}=\begin{bmatrix}E_{\mathrm{S}}&0&vt_{\mathrm{so}}\\\
0&2\mu_{\mathrm{QD}}+U_{\mathrm{QD}}+E_{\mathrm{S}}&-ut_{\mathrm{so}}\\\
vt_{\mathrm{so}}&-ut_{\mathrm{so}}&\mu_{\mathrm{QD}}-E_{\mathrm{Z}}+E_{\downarrow}\\\
\end{bmatrix}$

and

$H_{odd}=\begin{bmatrix}E_{\downarrow}&ut&0\\\
ut&\mu_{\mathrm{QD}}-E_{\mathrm{Z}}+E_{\mathrm{S}}&vt\\\
0&vt&2\mu_{\mathrm{QD}}+U_{\mathrm{QD}}+E_{\downarrow}\\\ \end{bmatrix}$

Here $E_{\mathrm{S}}$ and $E_{\downarrow}$ are the ABS singlet and doublet
energies respectively. The full many-body matrix describing the system is
given by the following block-diagonal matrix:

$H_{total}=\begin{bmatrix}H_{even}&0\\\ 0&H_{odd}\\\ \end{bmatrix}$ (10)

The ground state of the system corresponds to the eigenstate of the lowest
eigenenergy of Eq. 10. Fig. 3b was computed for $U_{\mathrm{ABS}}=0.03$,
$U_{\mathrm{QD}}=1$, $\Gamma=0.3$, $E_{\mathrm{Z}}=0.7$, $t=0.15$,
$t_{\mathrm{SO}}=0.01$.

###  VI Fig. 3 DC Transport and RF reflectometry comparison

In Fig. S6 we compare the charge stability diagram of Fig. 3a as measured in
RF reflectometry and DC transport. Most of the features seen in Fig. S6a are
also seen in Fig. S6b. In $V_{\mathrm{RF}}^{\mathrm{R}}$, some features are
more visible than in DC transport. While there is only a finite current when
the QD undergoes Andreev reflection or hybridizes with the ABS, there is
always a dip in $V_{\mathrm{RF}}^{\mathrm{R}}$ when the normal lead and QD are
on resonance. This can be explained by the excess dissipation known as
”Sisyphus resistance”, which originates from the QD level and Fermi level of
the normal lead being detuned with an AC voltage [27]. We attribute the
measured current at zero-bias in Fig. S6b to a finite voltage offset.

Figure S6: Comparison of DC transport and RF reflectrometry of Fig. 3. at
$V^{\mathrm{R}}_{\mathrm{bias}}=0$. a. Amplitude of the reflected RF signal of
the right lead, $V_{\mathrm{RF}}^{\mathrm{R}}$ for varying QD plunger gate
voltage, $V_{\mathrm{QD}}$, and $V_{\textrm{H}}$. a. Current measured at the
right lead, $I$, for varying QD plunger gate voltage, $V_{\mathrm{QD}}$, and
$V_{\textrm{H}}$.

###  VII Charge Sensor SNR

To estimate the SNR of the charge sensor, we measure the in-phase response and
quadrature, I and Q, of the left resonator for the ABS of Fig. 2 at
$B=$100\text{\,}\mathrm{m}\mathrm{T}$$. We measure the I and Q signals over
time at different $V_{\textrm{H}}$ values and plot the result in histograms,
as shown in Fig. S7a. For each histogram, we compute the average IQ response
$\mu_{i}$ and the standard deviation $\sigma_{i}$. Then, we define the SNR for
two charge states as:

$\mathrm{SNR}_{\mathrm{i,j}}=\frac{|\mu_{i}-\mu_{j}|}{\sigma_{i}+\sigma_{j}}$

See Supplemental Material of Ref. [63] for details. The red and purple lines
in Fig. S7a show $|\mu_{i}-\mu_{j}|$ for the 0 and 1, and 1 and 2 occupations
of the ABS. Dividing by the sum of the standard deviations results in the SNR.
Fig. S7b shows SNR calculated for varying integration time,
$t_{\mathrm{int}}$. Fig. S7c shows SNR calculated for varying RF output power.
It reaches a maximum of $39.5$ for -26 dBm for the SNR of the 0-like singlet
and 1 occupations. Fig. S7d shows SNR calculated for varying tunnel gate
voltage $V_{\mathrm{LO}}$. We fit the resulting Coulomb peak and extract the
ratio of peak height and full width at half maximum. We see that a sharper
Coulomb peak results in a higher SNR.

Figure S7: Charge Sensor SNR measurements for a single ABS. a. Example
histograms of the in-phase response and quadrature, I and Q of the left
resonator, for the 0-like even, 1 and 2-like even occupations of the ABS. The
colored markers indicate the charge state of the ABS that corresponds to each
histogram. The red and purple lines show the distance between the histograms
that we use to calculate the SNR. b. The SNR calculated from IQ-histograms for
the 0-like singlet and doublet states (red), and 2-like singlet and doublet
states (purple) for varying integration time $t_{\mathrm{int}}$. The vertical
black line indicates, $t_{\mathrm{int}}=$9.3\text{\,}\mathrm{m}\mathrm{s}$$,
which is the integration time used in the main text. The RF out signal had a
power of $-30\text{\,}~{}\mathrm{d}\mathrm{B}\mathrm{m}$ for every
$t_{\mathrm{int}}$. c. Same as panel b., for varying power of the RF input
signal and at fixed $t_{\mathrm{int}}=$9.3\text{\,}\mathrm{m}\mathrm{s}$$. d.
Same as panel c., for varying shape of the CS Coulomb resonance at fixed
$t_{\mathrm{int}}=$9.3\text{\,}\mathrm{ms}$$ and power
$-30\text{\,}~{}\mathrm{d}\mathrm{B}\mathrm{m}$. Each datapoint is taken for a
different gate voltage $V_{\mathrm{LO}}$, resulting in Coulomb peaks of
different width. We present the SNR versus Coulomb peak height, divided by the
full width at half maximum.


# A Radon-Nikodym theorem for monotone measures

Yao Ouyang
Faculty of Science, Huzhou Teacher’s College, Huzhou, Zhejiang 313000,
People’s Republic of China
Jun Li
State Key Laboratory of Media Convergence and Communication,
Communication University of China, Beijing 100024, China
E-mail:oyy@zjhu.edu.cn(Y. Ouyang)Corresponding author.
E-mail:lijun@cuc.edu.cn(J. Li)

###### Abstract

A version of Radon-Nikodym theorem for the Choquet integral w.r.t. monotone
measures is proved. Without any presumptive condition, we obtain a necessary
and sufficient condition for the ordered pair $(\mu,\nu)$ of finite monotone
measures to have the so-called Radon-Nikodym property related to a nonnegative
measurable function $f$. If $\nu$ is null-continuous and weakly null-additive,
then $f$ is uniquely determined almost everywhere by $\nu$ and thus is called
the Radon-Nikodym derivative of $\mu$ w.r.t. $\nu$. For $\sigma$-finite
monotone measures, a Radon-Nikodym type theorem is also obtained under the
assumption that the monotone measures are lower continuous and null-additive.

Keywords:Radon-Nikodym theorem; Monotone measure; Choquet integral; lower
continuous; null-additive

##  1 Introduction

Suppose that $\nu$ is a $\sigma$-additive measure and $f$ is a nonnegative
integrable function. The measure $\mu$ defined by

$\mu(A)=\int_{A}fd\nu$

for all measurable sets $A$ is said to be the indefinite integral of $f$
w.r.t. $\nu$. In this case, $\mu$ is absolutely continuous w.r.t. $\nu$. Under
what conditions a measure can be expressed as the indefinite integral w.r.t.
another measure is quite interesting. This pertains to the scope of the Radon-
Nikodym theorem. Radon-Nikodym theorem, one of the most important theorems in
measure theory, states that $\mu$ is the indefinite integral w.r.t. $\nu$ if
and only if $\mu$ is absolutely continuous w.r.t. $\nu$, see Halmos [9] for
example. We note that the Radon-Nikodym theorem has various proofs and all
these proofs are highly dependent on the $\sigma$-additivity of measures.

When one of the measures is only finitely additive, the Radon-Nikodym theorem
does not hold in general. Since in this case, the Hahn decomposition does not
hold in general and the implications “$\nu(A)=0\Rightarrow\mu(A)=0$” and
“$\nu(A_{n})\to 0\Rightarrow\mu(A_{n})\to 0$” are not equivalent. Various
conditions [2, 3, 14] have been derived in the literature for the validity of
finitely additive measures-based Radon-Nikodym theorem. For example, in [2]
the Radon-Nikodym theorem was proved under absolute continuity and a property
called Hahn separation (a variant of Hahn decomposition).

Graf [7] proved a Radon-Nikodym theorem for the Choquet integral w.r.t.
capacities (lower continuous subadditive monotone measure), while Nguyen et
al. [15, 16] investigated a Radon-Nikodym theorem for $\sigma$-subadditive
monotone measures. Greco [8] (see also [4]) obtained necessary and sufficient
conditions of this theorem for null-additive monotone measures. Roughly
speaking, these conditions include a variant of Hahn decomposition and some
other conditions. We also note that Rébillé [20] discussed the superior Radon-
Nikodym derivative of a set function w.r.t. a $\sigma$-additive measure.

In this paper, a new version of Radon-Nikodym theorem for the Choquet integral
is proved. It should be stressed that our result generalizes the corresponding
ones in [7, 8, 15]. Concretely, we introduce the concept of decomposition
property of monotone measures in Section 3. This property concerns an ordered
pair $(\mu,\nu)$ of monotone measures and a decreasing family
$\\{A_{\alpha}\\}_{\alpha\in\mathbb{Q}_{+}}$ of measurable sets and is a
natural generalization of the Hahn decomposition for $\sigma$-additive
measures. The decomposition property together with
$\lim\limits_{\alpha\to\infty}\mu(A_{\alpha})\vee\nu(A_{\alpha})=0$ is
demonstrated to be the necessary and sufficient conditions for $(\mu,\nu)$ to
have Radon-Nikodym property based on the Choquet integral (i.e., there is a
nonnegative measurable function $f$ such that $\mu(E)=\int_{E}fd\nu$ for each
measurable set $E$), where $\mu,\nu$ are finite monotone measures. This result
is obtained without any presumptive condition other than the monotonicity of
set functions, thus it is a generalization of the results of Greco [8] and
Nguyen et al. [16]. When $\nu$ is further weakly null-additive (which is
weaker than subadditive) and null-continuous (which is implied by lower
continuous), then the function $f$ is unique a.e.[$\nu$] and is called the
Radon-Nikodym derivative of $\mu$ w.r.t. $\nu$. Thus, Graf’s result is also
generalized. The Radon-Nikodym theorem for $\sigma$-finite monotone measures
are considered in Section 4. The existence and uniqueness of the Radon-Nikodym
derivative is obtained when $\mu,\nu$ are lower continuous and $\nu$ is
further null-additive.

##  2 Preliminaries

Let $(U,{\mathcal{U}})$ denote a measurable space, that is, a nonempty set $U$
equipped with a $\sigma$-algebra ${\mathcal{U}}$ of subsets of $U$. A subset
$A$ of $U$ is called measurable (w.r.t. ${\mathcal{U}}$) if
$A\in{\mathcal{U}}$. A nonnegative extended real-valued function $f\colon
U\to\overline{\mathbb{R}}_{+}$ is called measurable if for each
$\alpha\in[0,+\infty]$, $\\{f\geq\alpha\\}\in{\mathcal{U}}$ (here
$\\{f\geq\alpha\\}$ is the abbreviation for $\\{t\in
U\,|\,f(t)\geq\alpha\\}$).

######  Definition 2.1.

A set function $\mu:{\mathcal{U}}\rightarrow\overline{\mathbb{R}}_{+}$ is
called a monotone measure if it satisfies the following two conditions:

(i) $\mu(\emptyset)=0$; (vanishing at $\emptyset$)

(ii) $\mu(A)\leq\mu(B)$ whenever $A\subset B$ and $A,B\in{\mathcal{U}}$.
(monotonicity)

The triple $(U,{\mathcal{U}},\mu)$ is called a _monotone measure space_.

A monotone measure $\mu$ on $(U,{\mathcal{U}})$ is said to be (i) _finite_ if
$\mu(U)<\infty$; (ii) _$\sigma$ -finite_ if there is
$\\{U_{n}\\}_{n=1}^{\infty}\subset{\mathcal{U}}$ with $U_{n}\nearrow U$
(_i.e._ , $U_{1}\subset U_{2}\subset\cdots\subset U_{n}\subset\cdots$ and
$\bigcup\limits_{n=1}^{\infty}U_{n}=U$) such that $\mu(U_{n})<\infty$ for each
$n$.

Let $f$ be a nonnegative measurable function, $\nu$ be a monotone measure and
$A$ be a measurable set. The Choquet integral of $f$ w.r.t. $\nu$ is defined
as follows, see [5, 6, 19].

######  Definition 2.2.

The Choquet integral of $f$ w.r.t. $\nu$ on $A$ is given by

$\int_{A}fd\nu=\int_{0}^{\infty}\nu(\\{f\geq\alpha\\}\cap A)d\alpha,$

where the integral on the right side is the improper Riemann integral.

When $A=U$, we write $\int fd\nu$ instead of $\int_{U}fd\nu$. If $\int
fd\nu<\infty$, then $f$ is called Choquet integrable w.r.t. $\nu$ on $U$. When
$\nu$ is a $\sigma$-additive measure, the Choquet integral coincides with the
Lebesgue integral.

Throughout this paper, unless otherwise stated, all the considered integrals
are assumed to be the Choquet integrals. The following are some basic
properties of the Choquet integrals ([6, 19, 25]):

######  Proposition 2.3.

Let $(U,{\mathcal{U}},\nu)$ be a monotone measure space and $f,g$ be
nonnegative measurable functions. Then

(i) $\int_{A}fd\nu=0$ whenever $\nu(A)=0$;

(ii) $f\leq g$ implies $\int fd\nu\leq\int gd\nu$; (monotonicity)

(iii) $\int cfd\nu=c\int fd\nu$ for any constant $c\geq 0$; (homogeneity)

(iv) $\int_{A}\chi_{A}d\nu=\nu(A),\forall\,A\in{\mathcal{U}}$, where
$\chi_{A}$ denotes the characteristic function of $A$;

(v) $\int_{A}fd\nu=\int f\chi_{A}d\nu$;

(vi) $\int_{A}fd\nu=\lim\limits_{n\to\infty}\int_{A}(f\wedge n)d\nu$.

Proof. We only give the proof of (vi). For any $A\in{\mathcal{U}}$,

$\displaystyle\int_{A}fd\nu$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\nu(A\cap\\{f\geq\alpha\\})d\alpha=\lim_{n\to\infty}\int_{0}^{n}\nu(A\cap\\{f\geq\alpha\\})d\alpha$
$\displaystyle=$
$\displaystyle\lim_{n\to\infty}\int_{0}^{n}\nu(A\cap\\{f\wedge
n\geq\alpha\\})d\alpha$ $\displaystyle=$
$\displaystyle\lim_{n\to\infty}\left(\int_{0}^{n}\nu(A\cap\\{f\wedge
n\geq\alpha\\})d\alpha+\int_{n}^{\infty}\nu(A\cap\\{f\wedge
n\geq\alpha\\})d\alpha\right)$ $\displaystyle=$
$\displaystyle\lim_{n\to\infty}\int_{0}^{\infty}\nu(A\cap\\{f\wedge
n\geq\alpha\\})d\alpha=\lim_{n\to\infty}\int_{A}(f\wedge n)d\nu.$

$\Box$

Two functions $f,g$ on $U$ are said to be _comonotone_ if for any
$t_{1},t_{2}\in U$, $(f(t_{1})-f(t_{2}))(g(t_{1})-g(t_{2}))\geq 0$. The
following proposition is known as _comonotonic additivity_ of Choquet
integral, which is a distinguishing feature of the Choquet integral, see [6,
21].

######  Proposition 2.4.

Let $(U,{\mathcal{U}},\nu)$ be a monotone measure space and $f,g$ be
nonnegative measurable functions. If $f$ and $g$ are comonotone, then

$\int(f+g)d\nu=\int fd\nu+\int gd\nu.$

Note that two increasing (decreasing, resp.) functions are comonotone, and a
constant function $c$ is comonotone with arbitrary functions. Moreover, for
any function $f$ and any constant $c$, $(f-c)\vee 0$ and $f\wedge c$ are
comonotone, where $(f\vee c)(t)=\max\\{f(t),c\\}$ and $(f\wedge
c)(t)=\min\\{f(t),c\\}$.

##  3 Radon-Nikodym theorem for finite monotone measures

In this section we present a new version of Radon-Nikodym theorem for finite
monotone measures. To do this, we introduce the following concept of
decomposition property relating to an ordered pair of monotone measures.

######  Definition 3.1.

Let $\mu,\nu$ be two monotone measures on $(U,{\mathcal{U}})$. The ordered
pair $(\mu,\nu)$ is said to have _decomposition property_ if there is a
decreasing family $\\{A_{\alpha}\\}_{\alpha\in\mathbb{Q}^{+}}$ of measurable
sets with $A_{0}=U$ such that

$\displaystyle\alpha\Big{(}\nu(A\cap A_{\alpha})-\nu(A\cap A_{\beta})\Big{)}$
$\displaystyle\leq$ $\displaystyle\mu(A\cap A_{\alpha})-\mu(A\cap A_{\beta})$
(1) $\displaystyle\leq$ $\displaystyle\beta\Big{(}\nu(A\cap
A_{\alpha})-\nu(A\cap A_{\beta})\Big{)}$ (2)

holds for any $A\in{\mathcal{U}}$ with finite measures for $\mu$ and $\nu$,
and any $\alpha,\beta\in\mathbb{Q}^{+}$ with $\alpha<\beta$, where
$\mathbb{Q}^{+}$ is the set of all nonnegative rational numbers.

######  Example 3.2.

For any $\sigma$-additive finite measures $\mu,\nu$, the ordered pair
$(\mu,\nu)$ has decomposition property w.r.t.
$\\{A_{\alpha}\\}_{\alpha\in\mathbb{Q}^{+}}$, where $(A_{\alpha},A^{\rm
c}_{\alpha})$ is a Hahn decomposition of the signed measure $\mu-\alpha\nu$
(see Remark 3.8 for detail). This is why we call the ordered pair $(\mu,\nu)$
having decomposition property if $\mu,\nu$ satisfy inequalities (1) and (2).

######  Lemma 3.3.

Let $(\mu,\nu)$ have decomposition property w.r.t.
$\\{A_{\alpha}\\}_{\alpha\in\mathbb{Q}^{+}}$. If
$\lim\limits_{\alpha\to\infty}\mu(A_{\alpha})\vee\nu(A_{\alpha})=0$, then
$\lim\limits_{\alpha\to\infty}\alpha\nu(A_{\alpha})=0$.

Proof. We can assume that $\mu(A_{\alpha})\vee\nu(A_{\alpha})<\infty$. From
the first inequality in Definition 3.1, we get

$\alpha\Big{(}\nu(A_{\alpha})-\nu(A_{\beta})\Big{)}\leq\mu(A_{\alpha})-\mu(A_{\beta})$

holds for any $\alpha<\beta$, and hence we have
$\alpha\nu(A_{\alpha})\leq\mu(A_{\alpha})$ by letting $\beta\to\infty$ as
$\lim\limits_{\beta\to\infty}\mu(A_{\beta})\vee\nu(A_{\beta})=0$. Thus we also
have $\alpha\nu(A_{\alpha})$ tends to $0$ whenever $\alpha$ tends to $\infty$.
$\Box$

Now we show our main result — a version of Radon-Nikodym theorem for finite
monotone measures.

######  Theorem 3.4.

Let $\mu,\nu$ be two finite monotone measures on $(U,{\mathcal{U}})$. Then the
following two assertions are equivalent:

(i) The ordered pair $(\mu,\nu)$ has _Radon-Nikodym property_ , i.e., there is
a nonnegative measurable function $f\colon U\to\overline{\mathbb{R}}_{+}$ such
that

$\mu(A)=\int_{A}fd\nu,\ \ \forall\,A\in{\mathcal{U}}.$ (3)

(ii) The ordered pair $(\mu,\nu)$ has decomposition property w.r.t. a sets
system $\\{A_{\alpha}\\}_{\alpha\in\mathbb{Q}^{+}}$ and
$\lim\limits_{\alpha\to\infty}\mu(A_{\alpha})\vee\nu(A_{\alpha})=0$.

Proof. (i)$\Longrightarrow$ (ii). Suppose there is a nonnegative measurable
function $f$ such that Eq. (3) holds, then $f$ is Choquet integrable w.r.t.
$\nu$ on $U$, i.e., $\int fd\nu=\mu(U)<\infty$.

First we show that $(\mu,\nu)$ has decomposition property. Put
$A_{\alpha}=\\{f\geq\alpha\\}$, then $\\{A_{\alpha}\\}$ is decreasing and
$A_{0}=U$. For any $A\in{\mathcal{U}}$ with finite measure (i.e.,
$\mu(A)\vee\nu(A)<\infty$), and any $\alpha<\beta$ it holds

$\displaystyle\mu(A\cap A_{\alpha})-\mu(A\cap A_{\beta})$ $\displaystyle=$
$\displaystyle\int_{A\cap A_{\alpha}}fd\nu-\int_{A\cap A_{\beta}}fd\nu$
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}\Big{(}\nu(A\cap
A_{\alpha}\cap A_{t})-\nu(A\cap A_{\beta}\cap A_{t})\Big{)}dt$
$\displaystyle\geq$ $\displaystyle\int_{0}^{\alpha}\Big{(}\nu(A\cap
A_{\alpha})-\nu(A\cap A_{\beta})\Big{)}dt$ $\displaystyle=$
$\displaystyle\alpha\Big{(}\nu(A\cap A_{\alpha})-\nu(A\cap A_{\beta})\Big{)}.$

On the other hand, we have

$\displaystyle\mu(A\cap A_{\alpha})-\mu(A\cap A_{\beta})$ $\displaystyle=$
$\displaystyle\int_{0}^{\beta}\Big{(}\nu(A\cap A_{\alpha}\cap A_{t})-\nu(A\cap
A_{\beta})\Big{)}dt$ $\displaystyle+\int_{\beta}^{\infty}\Big{(}\nu(A\cap
A_{t})-\nu(A\cap A_{t})\Big{)}dt$ $\displaystyle\leq$
$\displaystyle\int_{0}^{\beta}\Big{(}\nu(A\cap A_{\alpha})-\nu(A\cap
A_{\beta})\Big{)}dt$ $\displaystyle=$ $\displaystyle\beta\Big{(}\nu(A\cap
A_{\alpha})-\nu(A\cap A_{\beta})\Big{)}.$

The assertion $\lim\limits_{\alpha\to\infty}\nu(A_{\alpha})=0$ follows from
$\alpha\nu(A_{\alpha})\leq\int_{A_{\alpha}}fd\nu=\mu(A_{\alpha})<\infty$.

Since $f=((f-n)\vee 0)+(f\wedge n)$ and $f=((f-n)\vee 0)$ and $(f\wedge n)$
are comonotone, then

$\int fd\nu=\int((f-n)\vee 0)d\nu+\int(f\wedge n)d\nu$

holds for each $n$. Therefore, from $\int
fd\nu=\lim\limits_{n\to\infty}\int(f\wedge n)d\nu$ and noting that $\int
fd\nu<\infty$, we conclude that

$\lim\limits_{n\to\infty}\int((f-n)\vee 0)d\nu=0.$

Also,

$\displaystyle\mu(A)=\int_{A}fd\nu$ $\displaystyle=$
$\displaystyle\int_{A}((f-n)\vee 0)d\nu+\int_{A}(f\wedge n)d\nu$
$\displaystyle\leq$ $\displaystyle\int((f-n)\vee 0)d\nu+n\nu(A)$

for each $A\in\mathcal{U}$. Specifically,

$\mu(A_{n})\leq\int((f-n)\vee 0)d\nu+n\nu(A_{n})$

for each $n$, it follows that $\lim\limits_{n\to\infty}\mu(A_{n})=0$.
Therefore, $\lim\limits_{\alpha\to\infty}\mu(A_{\alpha})=0$ as
$\\{A_{\alpha}\\}_{\alpha\geq 0}$ is a decreasing family of measurable sets.

(ii)$\Longrightarrow$(i). Suppose that $(\mu,\nu)$ has decomposition property
and $\\{A_{\alpha}\\}_{\alpha\in\mathbb{Q}^{+}}$ is the corresponding sets
system satisfying
$\lim\limits_{\alpha\to\infty}\mu(A_{\alpha})\vee\nu(A_{\alpha})=0$. We show
that there is a nonnegative measurable function $f\colon
U\to\overline{\mathbb{R}}_{+}$ such that Eq. (3) holds.

Define $f\colon U\to[0,\infty]$ as

$f(x)=\sup\\{\alpha\,|\,x\in A_{\alpha}\\}.$ (4)

Then $f$ is measurable and $\mu(\\{f=\infty\\})=0$ as
$\\{f=\infty\\}=\bigcap\limits_{\alpha\in\mathbb{N}}A_{\alpha}$.

For each positive integer $n$, define

$f_{n}(x)=\left\\{\begin{array}[]{ll}\frac{k-1}{2^{n}},&\mbox{if\,\,}x\in
A_{\frac{k-1}{2^{n}}}\setminus A_{\frac{k}{2^{n}}},k=1,2,\cdots,n2^{n},\\\
n,&\mbox{if\,\,}x\in A_{n}.\end{array}\right.$

Then $\\{f_{n}\\}_{n\in\mathbb{N}}$ is an increasing sequence and $f\wedge
n-\frac{1}{2^{n}}\leq f_{n}\leq f$. Since $f_{n}$ can be rewritten as

$f_{n}=\frac{1}{2^{n}}\sum_{k=1}^{n\cdot 2^{n}}\chi_{A_{\frac{k}{2^{n}}}},$

for any given $A\in{\mathcal{U}}$, we have

$\displaystyle\int_{A}f_{n}d\nu=\frac{1}{2^{n}}\sum_{k=1}^{n\cdot
2^{n}}\nu(A\cap A_{\frac{k}{2^{n}}})$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{n\cdot 2^{n}-1}\frac{k}{2^{n}}\Big{(}\nu(A\cap
A_{\frac{k}{2^{n}}})-\nu(A\cap A_{\frac{k+1}{2^{n}}})\Big{)}+n\nu(A\cap
A_{n})$ $\displaystyle\leq$ $\displaystyle\sum_{k=1}^{n\cdot
2^{n}-1}\Big{(}\mu(A\cap A_{\frac{k}{2^{n}}})-\mu(A\cap
A_{\frac{k+1}{2^{n}}})\Big{)}+n\nu(A\cap A_{n})$ $\displaystyle=$
$\displaystyle\mu(A\cap A_{\frac{1}{2^{n}}})-\mu(A\cap A_{n})+n\nu(A\cap
A_{n}).$

It follows from the assumption and Lemma 3.3 that

$\lim\limits_{n\to\infty}\int_{A}f_{n}d\nu\leq\lim\limits_{n\to\infty}\mu(A\cap
A_{\frac{1}{2^{n}}})\leq\mu(A).$

The inequality $f\wedge n\leq f_{n}+\frac{1}{2^{n}}$ implies that

$\int_{A}(f\wedge
n)d\nu\leq\int_{A}f_{n}d\nu+\int_{A}\frac{1}{2^{n}}d\nu=\int_{A}f_{n}d\nu+\frac{1}{2^{n}}\nu(A).$

By virtue of Proposition 2(vi) we get

$\int_{A}fd\nu=\lim\limits_{n\to\infty}\int_{A}(f\wedge
n)d\nu\leq\lim\limits_{n\to\infty}\int_{A}f_{n}d\nu\leq\mu(A).$

On the other hand,

$\displaystyle\int_{A}f_{n}d\nu=\frac{1}{2^{n}}\sum_{k=1}^{n\cdot
2^{n}}\nu(A\cap A_{\frac{k}{2^{n}}})$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{n\cdot 2^{n}-1}\frac{k+1}{2^{n}}\Big{(}\nu(A\cap
A_{\frac{k}{2^{n}}})-\nu(A\cap A_{\frac{k+1}{2^{n}}})\Big{)}$ $\displaystyle\
\ \ \ \ \ \ \ \ +\ (n+\frac{1}{2^{n}})\nu(A\cap
A_{n})-\frac{1}{2^{n}}\nu(A\cap A_{\frac{1}{2^{n}}})$ $\displaystyle\geq$
$\displaystyle\sum_{k=1}^{n\cdot 2^{n}-1}\Big{(}\mu(A\cap
A_{\frac{k}{2^{n}}})-\mu(A\cap A_{\frac{k+1}{2^{n}}})\Big{)}$ $\displaystyle\
\ \ \ \ \ \ \ \ +\ (n+\frac{1}{2^{n}})\nu(A\cap
A_{n})-\frac{1}{2^{n}}\nu(A\cap A_{\frac{1}{2^{n}}})$ $\displaystyle\geq$
$\displaystyle\mu(A\cap A_{\frac{1}{2^{n}}})-\mu(A\cap
A_{n})+(n+\frac{1}{2^{n}})\nu(A\cap A_{n})-\frac{1}{2^{n}}\nu(A\cap
A_{\frac{1}{2^{n}}}).$

By the decomposition property we have

$\displaystyle\mu(A)-\mu(A\cap A_{\frac{1}{2^{n}}})$ $\displaystyle=$
$\displaystyle\mu(A\cap A_{0})-\mu(A\cap A_{\frac{1}{2^{n}}})$
$\displaystyle\leq$ $\displaystyle\frac{1}{2^{n}}(\nu(A\cap A_{0})-\nu(A\cap
A_{\frac{1}{2^{n}}}))\to 0\,(n\to\infty),$

_i.e._ , $\mu(A\cap A_{\frac{1}{2^{n}}})\to\mu(A)\,(n\to\infty)$. Since both
$\mu(A_{n})$ and $n\nu(A_{n})$ tend to $0$ when $n\to\infty$, it then holds
that

$\int_{A}fd\nu\geq\lim\limits_{n\to\infty}\int_{A}f_{n}d\nu\geq\mu(A)$

as $f\geq f_{n}$ for each $n$. Thus we reach Eq. (3),

$\mu(A)=\int_{A}fd\nu,\ \ \forall\,A\in{\mathcal{U}}.$

The proof is complete. $\Box$

The Radon-Nikodym theorem for classical measures concerns the absolute
continuity of measures [9]. For monotone measures, there are various types of
absolute continuity (see [12, 17, 23, 24]). Let $\mu,\nu$ be two monotone
measures on $(U,{\mathcal{U}})$. (1) If for any $A\in{\mathcal{U}}$,
$\nu(A)=0$ implies $\mu(A)=0$, then we say that $\mu$ is absolutely continuous
w.r.t. $\nu$ and denoted by $\mu\ll\nu$. (2) If for each $\epsilon>0$ there is
a $\delta>0$ such that $\mu(A)<\epsilon$ for all sets $A\in{\mathcal{U}}$
satisfying $\nu<\delta$, then we say that $\mu$ is strongly absolutely
continuous w.r.t. $\nu$ and denoted by $\mu\ll^{s}\nu$ ([12]).

Obviously, $\mu\ll^{s}\nu$ implies $\mu\ll\nu$, but the converse is not true.

Observe that Theorem 3.4(ii) implies $\mu\ll\nu$ and $\mu\ll^{s}\nu$. In fact,
assume $\nu(A)=0$. From the second inequality in Definition 3.1, we take
$\alpha=0,\beta>0$, then

$\mu(A)-\mu(A\cap A_{\beta})\leq\beta\Big{(}\nu(A)-\nu(A\cap
A_{\beta})\Big{)},$

which implies $\mu(A)-\mu(A\cap A_{\beta})=0$ for any $\beta>0$. Therefore,
$\mu(A)=0$ as $\lim_{\beta\to\infty}\mu(A_{\beta})=0$. Similarly,
$\mu\ll^{s}\nu$ is also true.

Thus, we obtain necessary conditions that the Radon-Nikodym theorem in
classical measure theory remains valid for the Choquet integral w.r.t.
monotone measures (see also [24]).

######  Corollary 3.5.

Let $\mu,\nu$ be two finite monotone measures on $(U,{\mathcal{U}})$. If there
is a nonnegative measurable function $f\colon U\to\overline{\mathbb{R}}_{+}$
such that Eq. (3) holds, i.e.,

$\mu(A)=\int_{A}fd\nu,\ \ \forall\,A\in{\mathcal{U}},$

then $\mu\ll\nu$ and $\mu\ll^{s}\nu$.

Note that the measurable function $f$ in Theorem 3.4 is not unique in general.

######  Example 3.6.

Let $U$ be the set of all positive integers, $\mathcal{U}$ the power set of
$U$ and

$A_{\alpha}=\left\\{\begin{array}[]{ll}U,&\quad\text{if}\
\alpha\in[0,1]\cap\mathbb{Q},\\\\[2.84526pt]
\\{2,4,6,\cdots\\},&\quad\text{if}\
\alpha\in(1,2]\cap\mathbb{Q},\\\\[2.84526pt] \emptyset,&\quad\text{if}\
\alpha\in(2,\infty)\cap\mathbb{Q}.\end{array}\right.$

Define

$\mu(A)=\nu(A)=\left\\{\begin{array}[]{ll}1,&\quad\text{if\,}A=U,\\\\[2.84526pt]
0,&\quad\text{otherwise.}\end{array}\right.$

It is routine to verify that $(\mu,\nu)$ has decomposition property w.r.t.
$\\{A_{\alpha}\\}_{\alpha\in\mathbb{Q}^{+}}$. The condition
$\lim\limits_{\alpha\to\infty}\mu(A_{\alpha})\vee\nu(A_{\alpha})=0$ is
obviously satisfied. By Theorem 3.4 there exists a nonnegative function $f$
such that $\mu(A)=\int_{A}fd\nu$ for each $A$. In fact

$f_{1}(x)=\sup\\{\alpha\ |\ x\in
A_{\alpha}\\}=\left\\{\begin{array}[]{ll}1,&\quad\text{if\, $x$ is
odd}\\\\[2.84526pt] 2,&\quad\text{if\, $x$ is even}\end{array}\right.$

is such a function. Note that $f_{2}$ defined by

$f_{2}(x)=\left\\{\begin{array}[]{ll}1,&\quad\text{if\, $x$ is
even}\\\\[2.84526pt] 2,&\quad\text{if\, $x$ is odd}\end{array}\right.$

also satisfies $\mu(A)=\int_{A}f_{2}d\nu$ (there are in fact infinitely many
such functions). Interestingly, $f_{1}$ and $f_{2}$ are different at every
point and thus $\nu(\\{f_{1}\neq f_{2}\\})=\nu(U)=1$.

To ensure the uniqueness of $f$, we have to impose some additional conditions.
Recall that a monotone measure $\mu$ is said to be (i) weakly null-additive
[25], if $\mu(A_{1}\cup A_{2})=0$ for any $A_{1},A_{2}\in{\mathcal{U}}$ with
$\mu(A_{1})=\mu(A_{2})=0$; (ii) null-continuous [1], if
$\mu(\bigcup_{n=1}^{\infty}A_{n})=0$ for every increasing sequence
$\\{A_{n}\\}_{n\in N}\subset{\mathcal{A}}$ such that
$\mu(A_{n})=0,n=1,2,\cdots.$ The monotone measure $\mu$ is both weakly null-
additive and null-continuous if and only if
$\mu(\bigcup_{n=1}^{\infty}A_{n})=0$ whenever
$\\{A_{n}\\}_{n\in\mathbb{N}}\subset{\mathcal{A}}$ and
$\mu(A_{n})=0,n=1,2,\cdots$, see [11]. Such a monotone measure $\mu$ is called
to have property ($\sigma$), i.e., the set of all $\mu$-null sets is a
$\sigma$-ideal, see [4].

######  Proposition 3.7.

Let $\mu,\nu$ be two monotone measures and $\nu$ is weakly null-additive and
null-continuous. If measurable functions $f,g\colon U\to[0,\infty]$ satisfy
Eq. (3), i.e.,

$\mu(A)=\int_{A}fd\nu=\int_{A}gd\nu,\ \ \forall\,A\in{\mathcal{U}},$

then $f=g\ a.e.[\nu]$ (i.e., $\nu(\\{f\neq g\\})=0$).

Proof. Let $A=\\{f>g\\}$ and $A_{n}=\\{f>g+\frac{1}{n}\\}$, then
$A_{n}\nearrow A$. We conclude that $\nu(A)=0$, otherwise there is some $n$
such that $\nu(A_{n})>0$ as $\nu$ is null-continuous. Then

$\displaystyle\int_{A_{n}}fd\nu\geq\int_{A_{n}}(g+\frac{1}{n})d\nu$
$\displaystyle=$ $\displaystyle\int_{A_{n}}gd\nu+\int_{A_{n}}\frac{1}{n}d\nu$
$\displaystyle=$
$\displaystyle\int_{A_{n}}gd\nu+\frac{1}{n}\nu(A_{n})>\int_{A_{n}}gd\nu,$

a contradiction. It holds similarly that $\nu(B)=0$, where $B=\\{g>f\\}$.
Since $\nu$ is weakly null-additive, we have $\nu(A\cup B)=0$. As a
consequence $\nu(\\{f\neq g\\})=\nu(A\cup B)=0.$ $\Box$

Note: We also obtain $f=g\ a.e.[\mu]$ (i.e., $\mu(\\{f\neq g\\})=0$).

Now by using Proposition 3.7 we can propose the concept of Radon-Nikodym
derivative for monotone measures. In Theorem 3.4, we consider that $\nu$ is
weakly null-additive and null-continuous (i.e., $\nu$ has property
($\sigma$)), then the measurable function $f$ on $U$ for which Eq. (3) holds
is called a Radon-Nikodym derivative (or Radon-Nikodym density) of $\mu$
w.r.t. $\nu$, and denoted by $\frac{d\mu}{d\nu}$ and Eq. (3) will be written
as $f=\frac{d\mu}{d\nu}$ or $d\mu=fd\nu$. Thus, the preceding Theorem 3.4
asserts that if the ordered pair $(\mu,\nu)$ satisfies the condition (ii) and
$\nu$ has property ($\sigma$), then any two Radon-Nikodym derivatives of $\mu$
w.r.t. $\nu$ are equal $a.e.\ [\nu]$, and so the notation $\frac{d\mu}{d\nu}$
is only ambiguous up to a $\nu$-null set.

We can discuss some properties of Radon-Nikodym derivative. For example,
suppose that $\mu,\lambda$ and $\nu$ are finite monotone measures on
$(U,{\mathcal{U}})$ and $\nu$ has property ($\sigma$), and $(\mu,\nu)$ and
$(\lambda,\nu)$ satisfy the condition (ii) in Theorem 3.4, respectively, then
$\frac{d\mu}{d\nu}$ and $\frac{d\lambda}{d\nu}$ exist. We write
$\frac{d\mu}{d\nu}=f$ and $\frac{d\lambda}{d\nu}=g$, if $f$ and $g$ are
comonotone, then we have

$\frac{d(\mu+\lambda)}{d\nu}=\frac{d\mu}{d\nu}+\frac{d\lambda}{d\nu}\ \
a.e.[\nu].$

######  Remark 3.8.

Let $\mu,\nu$ be $\sigma$-additive finite measures. For each nonnegative
rational number $\tau$ the signed measure $\mu-\tau\nu$ has a Hahn
decomposition, _i.e._ , there is a measurable set $A_{\tau}$ such that
$A_{\tau}$ is a positive set of $\mu-\tau\nu$ and $A^{\rm c}_{\tau}$ is a
negative set of $\mu-\tau\nu$. Since $A_{\tau}$ is also a positive set of
$\mu-\gamma\nu$ for any $\gamma\leq\tau$, without loss of generality we can
suppose that $\\{A_{\tau}\\}_{\tau\in\mathbb{Q}^{+}}$ is decreasing. If
$\tau=0$, then $U$ itself is a positive set of $\mu-\tau\nu=\mu$ and so
$A_{0}=U$. Let $\alpha<\beta$ be given. Since $A_{\alpha}$ is a positive set
of $\mu-\alpha\nu$, for any $A\in{\mathcal{U}}$,
$(\mu-\alpha\nu)(A\cap(A_{\alpha}\setminus A_{\beta}))\geq 0$, _i.e._ ,

$\alpha\Big{(}\nu(A\cap A_{\alpha})-\nu(A\cap A_{\beta})\Big{)}\leq\mu(A\cap
A_{\alpha})-\mu(A\cap A_{\beta}).$

Similarly, $A^{\rm c}_{\beta}$ is a negative set of $\mu-\beta\nu$ and hence
is a positive set of $\beta\nu-\mu$. For any $A\in{\mathcal{U}}$,
$(\beta\nu-\mu)(A\cap(A_{\alpha}\setminus A_{\beta}))\geq 0$, _i.e._ ,

$\mu(A\cap A_{\alpha})-\mu(A\cap A_{\beta})\leq\beta\Big{(}\nu(A\cap
A_{\alpha})-\nu(A\cap A_{\beta})\Big{)}.$

Thus $(\mu,\nu)$ has decomposition property and $(A_{\alpha},A^{\rm
c}_{\alpha})$ is a Hahn decomposition of the signed measure $\mu-\alpha\nu$.

If $\mu,\nu$ further satisfy $\mu\ll\nu$, then we have
$\lim\limits_{\alpha\to\infty}\nu(A_{\alpha})=0$ and hence
$\lim\limits_{\alpha\to\infty}\mu(A_{\alpha})=0$. In fact, if
$\lim\limits_{\alpha\to\infty}\nu(A_{\alpha})>0$, then
$(\mu-\alpha\nu)(A_{\alpha})\to-\infty$ when $\alpha\to\infty$, contradicting
with the fact that $A_{\alpha}$ is a positive set of $\mu-\alpha\nu$. On the
other hand, if
$\lim\limits_{\alpha\to\infty}\mu(A_{\alpha})\vee\nu(A_{\alpha})=0$, then
$\mu\ll\nu$. To see this, let $\nu(A)=0$ be given. For any $\beta>0$,

$\mu(A\cap A_{0})-\mu(A\cap A_{\beta})\leq\beta(\nu(A\cap A_{0})-\nu(A\cap
A_{\beta}))=0,$

which implies $\mu(A)=0$ as $A_{0}=U$ and $\mu(A_{\beta})\to
0\,(\beta\to\infty)$.

In conclusion, for two $\sigma$-additive finite measures $\mu,\nu$, the pair
$(\mu,\nu)$ has decomposition property, and $\mu\ll\nu$ if and only if
$\lim\limits_{\alpha\to\infty}\mu(A_{\alpha})\vee\nu(A_{\alpha})=0$.

As a special case of Theorem 3.4, we thus obtain a classical Radon-Nikodym
theorem.

######  Corollary 3.9.

Let $\mu,\nu$ be $\sigma$-additive finite measures on $(U,{\mathcal{U}})$.
There exists a nonnegative measurable function $f\colon
U\to\overline{\mathbb{R}}_{+}$ such that

$\mu(A)=\int_{A}fd\nu,\ \ \forall\,A\in{\mathcal{U}}$

if and only if $\mu\ll\nu$.

Let $\mu,\nu$ be bounded finitely additive measures such that for every
$\epsilon>0$ there exists a finite decomposition of $U$,
$\\{A_{1},\cdots,A_{n}\\}\subset{\mathcal{U}}$, satisfying
$\mu(A_{i})\vee\nu(A_{i})<\epsilon$. For such measures, Candeloro and
Martellotti proved in [3] that if $\mu\ll^{s}\nu$ and the set
$\\{(\mu(A),\nu(A))|A\in{\mathcal{U}}\\}$ is closed, then $(\mu,\nu)$
satisfies all requirements in Theorem 3.4(ii) and thus $(\mu,\nu)$ has Radon-
Nikodym property.

######  Remark 3.10.

There are several papers dealing with Radon-Nikodym theorem for monotone
measures (see, for example, Graf [7], Greco [8] and Nguyen et al. [15, 16]).
Graf obtained his result under the assumption that the monotone measures are
subadditive and lower continuous, while Nguyen et al. demanded the monotone
measures being $\sigma$-subadditive. Greco [8] (see also Theorem 1.2 in
Candeloro, Volčič [4]) posed an additional requirement that

$(*)\ \ \ \ \ \mu(S)=\nu(S)=0\Rightarrow\nu(A\cup S)=\nu(A),\,\ \forall
A\in{\mathcal{U}}.$

Specifically, under the condition $(*)$, Greco proved that there is a
nonnegative function $f$ such that

$\mu(A)=\int_{A}fd\nu,\,\ \forall A\in{\mathcal{U}}$

if and only if $(\mu,\nu)$ satisfies a strong decomposition property (S.D.P.
for short, see [7]) w.r.t. a sets system $\\{A_{\alpha}\\}$ and
$\lim\limits_{\alpha\to\infty}\mu(A_{\alpha})=0$. It is not difficult to see
that S.D.P. together with $\\{A_{\alpha}\\}$ and
$\lim\limits_{\alpha\to\infty}\mu(A_{\alpha})=0$ implies that $\mu\ll\nu$,
hence the condition $(*)$ says in fact that $\nu$ is null-additive.

In contrast to these results, our result have no additional requirements for
monotone measures other than a set of sufficient and necessary conditions.
Interestingly, Example 3.6 shows that the Radon-Nikodym theorem can hold even
for monotone measures without weakly null-additivity.

##  4 Radon-Nikodym theorem for $\sigma$-finite monotone measures

Before presenting a Radon-Nikodym theorem for $\sigma$-finite monotone
measures, we need some further properties of the Choquet integral.

Note that if $(\mu,\nu)$ has decomposition property w.r.t. a sets system
$\\{A_{\alpha}\\}_{\alpha\in\mathbb{Q}^{+}}$, then for any nonempty set
$V\in{\mathcal{U}}$ the ordered pair $(\mu|_{V},\nu|_{V})$ also has
decomposition property w.r.t. the system $\\{A_{\alpha}\cap
V\\}_{\alpha\in\mathbb{Q}^{+}}$.

A monotone measure $\mu$ is said to be (i) _lower continuous_ (or _continuous
from below_) if for any $\\{A_{n}\\}\subset{\mathcal{U}}$ with $A_{n}\nearrow
A$, it holds that $\mu(A)=\lim\limits_{n\to\infty}\mu(A_{n})$; (ii) _null-
additive_ if $\mu(A\cup N)=\mu(A)$ for any $A,N\in{\mathcal{U}}$ with
$\mu(N)=0$.

Obviously, lower continuity implies null-continuity and null-additivity
implies weak null-continuity, but not vice versa (see [11]).

######  Proposition 4.1.

[6, 22] Let $(U,{\mathcal{U}},\nu)$ be a monotone measure space and
$f,g,f_{n}\ (n=1,2,\cdots)$ be nonnegative measurable functions.

(i) If $f=g\ a.e.[\nu]$ and $\nu$ is null-additive, then $\int fd\nu=\int
gd\nu$.

(ii) If $\nu$ is lower continuous and $f_{n}\nearrow f$, then
$\lim\limits_{n\to\infty}\int f_{n}d\nu=\int fd\nu$.

In the following we suppose that the monotone measures $\mu,\nu$ are
$\sigma$-finite. Without loss of generality, we can assume that there is
$\\{U_{n}\\}_{n=1}^{\infty}\subset{\mathcal{U}}$ with $U_{n}\nearrow U$ such
that for every $n$, $\mu(U_{n})<\infty$ and $\nu(U_{n})<\infty$ hold
simultaneously.

######  Theorem 4.2.

Let $\mu,\nu$ be $\sigma$-finite and lower continuous and $\nu$ be null-
additive. If $(\mu,\nu)$ has decomposition property w.r.t. the system
$\\{A_{\alpha}\\}_{\alpha\in\mathbb{Q}^{+}}$ and
$\lim\limits_{\alpha\to\infty}\mu(A_{\alpha}\cap U_{n})\vee\nu(A_{\alpha}\cap
U_{n})=0$ for each $n$, then there is a nonnegative and finite a.e.[$\nu$]
measurable function $f$ such that

$\mu(A)=\int_{A}fd\nu,\ \ \forall\,A\in{\mathcal{U}}.$

In this case, $f$ is unique $a.e.[\nu]$.

Proof. For each $n$, $(\mu|_{U_{n}},\nu|_{U_{n}})$ also has decomposition
property w.r.t. the system $\\{A_{\alpha}\cap
U_{n}\\}_{\alpha\in\mathbb{Q}^{+}}$. Since
$\lim\limits_{\alpha\to\infty}\mu(A_{\alpha}\cap U_{n})\vee\nu(A_{\alpha}\cap
U_{n})=0$ also holds, according to Theorem 3.4, there is a nonnegative
measurable function $f_{n}$ on $U_{n}$ such that

$\mu|_{U_{n}}(E)=\int_{E}f_{n}d\nu|_{U_{n}}$

for each measurable subset $E$ of $U_{n}$. Equivalently, for each
$A\in{\mathcal{U}}$ we have

$\mu|_{U_{n}}(A_{n})=\int_{A_{n}}f_{n}d\nu|_{U_{n}},$

where $A_{n}=A\cap U_{n}$. By Proposition 3.7, for $n>m$ we have
$f_{n}|_{U_{m}}=f_{m}\ a.e.[\nu]$. Without loss of generality, by Proposition
4.1(i) we can assume that $f_{n}|_{U_{m}}=f_{m}$ as $\nu$ is null-additive.
Let $\tilde{f}_{n}(u)=f_{n}(u)$ for $u\in U_{n}$ and $\tilde{f}_{n}(u)=0$ for
$u\in U\setminus U_{n}$. Then

$\mu(A_{n})=\int_{A_{n}}\tilde{f}_{n}d\nu$

holds for each $n$. Note that the sequence $\\{\tilde{f}_{n}\\}$ is
nondecreasing and thus it is convergent everywhere. Denote
$f=\lim\limits_{n\to\infty}\tilde{f}_{n}$, then $f|_{U_{n}}=\tilde{f}_{n}$ for
each $n$. Thus

$\\{f=\infty\\}=\bigcup_{n=1}^{\infty}\Big{(}U_{n}\cap\\{f=\infty\\}\Big{)}=\bigcup_{n=1}^{\infty}\\{\tilde{f}_{n}=\infty\\},$

which implies that $f$ is finite a.e.[$\nu$] as $\nu$ is null-additive and
lower continuous. Moreover, $f|_{U_{n}}=\tilde{f}_{n}$ also implies that

$\mu(A_{n})=\int_{A_{n}}fd\nu.$

Since $A_{n}\nearrow A$, again by the lower continuity of $\mu,\nu$ we reach
the final conclusion

$\mu(A)=\lim_{n\to\infty}\mu(A_{n})=\lim_{n\to\infty}\int_{A_{n}}fd\nu=\int_{A}fd\nu.$

The uniqueness of $f$ follows from (i) of Proposition 4.1.

This completes the proof. $\Box$

######  Example 4.3.

Let $U$ be the set of natural numbers and ${\mathcal{U}}$ be the power set of
$U$. Let $\nu(A)=1$ for $A\neq\emptyset$, $\mu(A)=\max A$ if $A$ is finite and
$\mu(A)=\infty$ if $A$ is infinite. Then (i) $\nu$ is lower continuous and
null-additive; (ii)$\mu$ is lower continuous; (iii) $\nu$ is finite and $\mu$
is $\sigma$-finite. Let $A_{\alpha}=[\alpha,\infty)\cap U$ for each
$\alpha\in\mathbb{Q}^{+}$. Then we can verify that $(\mu,\nu)$ has
decomposition property w.r.t. the system
$\\{A_{\alpha}\\}_{\alpha\in\mathbb{Q}^{+}}$. It is easy to see that
$\mu(A)=\int_{A}fd\nu,\forall\,A\in{\mathcal{U}}$ for $f(x)=\sup\\{\alpha|x\in
A_{\alpha}\\}=x$. Note that
$\lim\limits_{\alpha\to\infty}\mu(A_{\alpha})=\infty$ and
$\lim\limits_{\alpha\to\infty}\nu(A_{\alpha})=1$. But
$\lim\limits_{\alpha\to\infty}\mu(A_{\alpha}\cap U_{n})\vee\nu(A_{\alpha}\cap
U_{n})=0$ for each $n$, where $U_{n}=\\{0,1,2,\cdots,n\\}$.

##  5 Concluding remarks

We have presented a version of the Radon-Nikodym theorem for finite monotone
measures (Theorem 3.4). As we have seen, we introduced the decomposition
property of the ordered pair $(\mu,\nu)$ of monotone measures (Definition 3.1)
and showed a necessary and sufficient condition that the Radon-Nikodym theorem
holds for the Choquet integral w.r.t. finite monotone measures. We point out
that our version has no additional conditions for finite monotone measures
(such as, subadditivity, or $\sigma$-subadditivity, or continuity from below,
etc.) other than monotonicity. The proof of this result is dependent on a
distinguished feature, namely, the comonotonic additivity, of the Choquet
integral. The uniqueness of Radon-Nikodym derivative and the case of
$\sigma$-finite monotone measures have also been considered.

Apart from the Choquet integral, there are other nonlinear integrals (the
concave integral [10] and pan-integral [25] for example) in the literature
that extend the Lebesgue integral. So, it would be an interesting topic to
explore the Radon-Nikodym theorem for these integrals. As these integrals lack
the comonotonic additivity, we need to seek new decomposition properties and
techniques. The known relationships among these integrals [10, 13, 18] may be
useful.

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