# The Mean: Axiomatics, Generalizations, Applications

John E. Gray<EMAIL_ADDRESS>[ Naval Surface Warfare Center
Dahlgren
Dahlgren Division
Sensor Technology Branch Q31
18444 Frontage Road Suite 327
Dahlgren, VA 22448-5161
Naval Surface Warfare Center Dahlgren  Andrew
<EMAIL_ADDRESS>[ Department of Mathematics and Statistics
Georgetown University
Washington DC 20057-1233
Georgetown University

###### Abstract

We present an axiomatic approach to the mean and discuss generalizations of
the mean, including one due to Kolmogorov based on the Weak Law of Large
Numbers. We offer examples and counterexamples, describe conventional and
unconventional uses of the mean in statistical mechanics, and resolve an
anomaly in quantum theory concerning apparent simultaneous coexistence of
means and variances of observables. These issues all arise from the familiar
definition of the mean.

60A05,

26E60, 62P35, 82B03, 94A17, 93A10, 81P10,

axiomatics for the mean,

median,

entropy,

Jaynes’ Maximum Entropy Principle,

weak mean,

###### keywords:

[class=AMS]

###### keywords:

and

##  1 Introduction

The most important number summarizing a data set is generally thought to be
the mean. Some have questioned its utility, comparing it unfavorably with the
median, the mode, the midrange. Capitalists and communists used to argue over
whether mean income or median income was the truer measure of citizen well-
being. For another example, see Kosko [9]. The mean is not robust against
outliers: it can be strongly influenced by a single observation. This is both
a strength and a weakness. Kosko objected that not only does a Cauchy random
variable not have a well-defined mean but the average of independent
identically distributed Cauchy random variables is itself a Cauchy variable
with the same distribution and thus averaging does not reduce variability at
all.

Investigators often pursue the quest for a single number or a small set of
numbers that capture the essence of a data set, make multiple data sets
comparable, and provide order to the world of data sets. As data sets get
larger and larger, thanks to the digital explosion, scrutiny of measures that
compress data becomes more important. Candidates, in addition to those
mentioned above, include entropy and various generalized means, but no one has
arrived at measures clearly superior to the mean and its associated measure,
the root mean squared deviation or standard deviation.

In work with sample data the mean is easy to understand, in contrast with
other notions from probability theory - such as independence, conditional
probability, and even probability itself. Some have argued (e.g., de Finetti
[2], Pollard [12], and Whittle [14]) that the mean is the fundamental notion
in probability theory and should occupy the central place in all treatments of
probability.

In this note we review some properties of the mean, consider some
generalizations for cases when the ordinary mean does not exist, and
investigate the significance of the mean in state space theory and quantum
mechanics.

We begin by axiomatizing the notion of sample mean. Along with familiar axioms
for symmetry, homogenity, and translation invariance, we introduce a
_condensation_ axiom that describes the result of replacing arbitrary values
by their sample mean. We then use the Strong Law of Large Numbers to arrive at
the familiar mathematical notion of mean, $E(X)$.

Thereafter we consider generalizations of the mean. These are not needed for
bounded or semi-bounded random variables, but really only for variables that
have heavy-tailed distributions on both right and left, with tails of similar
size. We consider what happens when a random variable is restricted to an
interval $[c-M,c+M]$ and $M$ is allowed to tend to infinity. We state a
theorem (Theorem 3.1) describing the different kinds of behavior possible and
provide examples of each. One generalization, which is due to Kolmogorov, is
what we have chosen to call the _weak mean_ , $E_{w}(X)$, and corresponds
precisely to validity of the Weak Law of Large Numbers. Yet another
generalization, the _doubly weak mean_ , $E_{ww}(X)$, applies to the Cauchy
distribution. We also discuss multipliers that can be applied to a variable
$X$ to finitize the mean in the spirit of Feynman and note the dangers of such
finitizations. Nonetheless, we recognize that attempts to scrutinize the
notion of mean in connection with the Cauchy distribution and other long-
tailed distributions are timely.

Turning to applications, we point out that the mean is a natural tool in state
space theory for the transition from deterministic models to statistical
models. We discuss entropy and observe that although it is regarded as a mean
it is very different from means arising from ordinary observables. We recall
Jaynes’ Maximum Entropy Principle, which seeks to maximize entropy subject to
given values of conventional means.

Lastly, we discuss the role the mean plays in quantum theory, and provide a
precise answer to the question of when the mean and variance exist for a
particular quantum state and a particular quantum observable.

The conclusion, implicit in this discussion, is that the mean is the paramount
measure, of great and wide utility, instructive even when it falls short.
There is little prospect of it losing its longtime preeminence.

##  2 Axiomatics for the Sample Mean and the Strong Law of Large Numbers

Prior to introducing probability measures, let us consider potential axioms
for the mean of a finite set. In this setting, with
$\mathcal{R}=(-\infty,\infty)$, the mean can be thought of as a family of
functions $\\{f_{n}\\}$ for $n\geq 1$ with
$f_{n}:\mathcal{R}^{n}\rightarrow\mathcal{R}$. Its properties include the
following:

M-1) (Homogeneity) $f_{n}(\lambda x_{1},\ldots,\lambda x_{n})=\lambda
f_{n}(x_{1},\ldots,x_{n})$ for all
$(x_{1},\ldots,x_{n})\in\mathcal{R}^{n}$ and all $\lambda\in\mathcal{R}$;

M-2) (Symmetry)
$f_{n}(x_{1},\ldots,x_{n})=f_{n}(x_{\sigma(1)},\ldots,x_{\sigma(n)})$ for all
permutations $\sigma$ of the set $\\{1,2,...,n\\}$;

M-3) (Translation Invariance)
$f_{n}(x_{1}+c,\ldots,x_{n}+c)=f_{n}(x_{1},\ldots,x_{n})+c$ for all
$(x_{1},\ldots,x_{n})\in\mathcal{R}^{n}$ and all c $\in\mathcal{R}$.

Other properties are the following:

(Positive Homogeneity) $f_{n}(\lambda x_{1},\ldots,\lambda x_{n})=\lambda
f_{n}(x_{1},\ldots,x_{n})$ for all
$(x_{1},\ldots,x_{n})\in\mathcal{R}^{n}$ and all $\lambda>0$;

(Nonnegativity) If for some $(x_{1},\ldots,x_{n})\mbox{ and
}(y_{1},\ldots,y_{n})\in\mathcal{R}^{n}$
$x_{1}\leq y_{1},\ldots,x_{n}\leq y_{n}$, then $f_{n}(x_{1},\ldots,x_{n})\leq
f_{n}(y_{1},\ldots,y_{n})$;

(Positivity) If for some $(x_{1},\ldots,x_{n})\mbox{ and
}(y_{1},\ldots,y_{n})\in\mathcal{R}^{n}$
$x_{1}\leq y_{1},\ldots,x_{n}\leq y_{n}$, and $x_{i}<y_{i}$ for some i, then
$f_{n}(x_{1},\ldots,x_{n})<f_{n}(y_{1},\ldots,y_{n})$.

(Strict Positivity) If for some $(x_{1},\ldots,x_{n})\mbox{ and
}(y_{1},\ldots,y_{n})\in\mathcal{R}^{n}$
$x_{i}<y_{i}$ for all i = 1, …, n, then
$f_{n}(x_{1},\ldots,x_{n})<f_{n}(y_{1},\ldots,y_{n})$.

(Additivity)
$f_{n}(x_{1}+y_{1},\ldots,x_{n}+y_{n})=f_{n}(x_{1},\ldots,x_{n})+f_{n}(y_{1},\ldots,y_{n})$
for all $(x_{1},\ldots,x_{n})\mbox{ and
}(y_{1},\ldots,y_{n})\in\mathcal{R}^{n}$;

The above axioms seem reasonable except for additivity. The measure should be
independent of units, thus homogeneous, and independent of the choice of zero
point, and a function of the set rather than the ordered set. In addition, to
capture characteristics of the data, ordering properties - nonnegativity and
perhaps positivity - are not unreasonable. However, additivity asserts a
relationship between the ordering of two data sets that survives reordering of
one set, and this seems much too restrictive.

Consider a rival measure to the mean, namely, the median. The median of a
finite data set $\\{x_{1},\ldots,x_{n}\\}$ is defined as the midmost of the
numbers when they are arranged in increasing order if $n$ is odd, and half the
sum of the two midmost numbers in such an arrangement if $n$ is even.

The median satisfies homogeneity, symmetry, translation invariance, and
nonnegativity. Furthermore, any fixed convex combination of the mean and the
median other than the median itself satisfies homogeneity, symmetry,
translation invariance, nonnegativity, positivity, and strict positivity.
Indeed, not only the median, but the maximum and the minimum of
$\\{x_{1},\ldots,x_{n}\\}$ (and other rank functions and convex combinations)
satisfy positive homogeneity, symmetry, translation invariance, and
nonnegativity.

###### Proposition 2.1

Let $f_{n}:\mathcal{R}^{n}\rightarrow\mathcal{R}$ be a function satisfying
homogeneity, symmetry, and translation invariance.

1) If n = 1, then $f_{1}(x)=x$ for all $x\in\mathcal{R}$.

2) If n = 2, then $f_{2}(x_{1},x_{2})=\frac{x_{1}+x_{2}}{2}$ for all
$(x_{1},x_{2})\in\mathcal{R}^{2}$.

Proof: Homogeneity implies that $f_{1}(0)=f_{2}(0,0)=0$. Translation
invariance then indicates that $f_{1}(x)=f_{1}(0+x)=f_{1}(0)+x=x$. When n = 2,

$\displaystyle f_{2}(a,b)$ $\displaystyle=$ $\displaystyle
f_{2}(-\frac{b-a}{2}+\frac{a+b}{2},\frac{b-a}{2}+\frac{a+b}{2})$
$\displaystyle=$ $\displaystyle
f_{2}(-\frac{b-a}{2},\frac{b-a}{2})+\frac{a+b}{2}$ $\displaystyle=$
$\displaystyle(\frac{b-a}{2})f_{2}(-1,1)+\frac{a+b}{2}.$

However, by homogeneity and symmetry $f_{2}(-1,1)=-f_{2}(1,-1)$
$=-f_{2}(-1,1)$, and $f_{2}(-1,1)=0$.

When n = 1 or 2, the median and the mean coincide. However, it is obvious that
they do not coincide in general when n is 3 or larger. Without the requirement
of additivity it is natural to inquire whether there is another suitable
property that will distinguish between the median and the mean. One property
that we consider and reject is that $f_{n}$ shall have continuous partial
derivatives.

###### Proposition 2.2

Let $f_{n}:\mathcal{R}^{n}\rightarrow\mathcal{R}$ be a function satisfying
homogeneity, symmetry, and translation invariance that has partial derivatives
at each point with the partial derivatives continuous at
$(0,\ldots,0)\in\mathcal{R}^{n}$. Then

$\displaystyle f_{n}(x_{1},\ldots,x_{n})=\frac{x_{1}+\ldots+x_{n}}{n}$

for all $(x_{1},\ldots,x_{n})\in\mathcal{R}^{n}$.

Proof: If we differentiate the equation $f_{n}(\lambda x_{1},\ldots,\lambda
x_{n})=\lambda f_{n}(x_{1},\ldots,x_{n})$ with respect to $x_{i}$, we obtain:

$\displaystyle\lambda\frac{\partial f_{n}}{\partial x_{i}}(\lambda
x_{1},\ldots,\lambda x_{n})$ $\displaystyle=$
$\displaystyle\lambda\frac{\partial f_{n}}{\partial
x_{i}}(x_{1},\ldots,x_{n}).$

Cancelling $\lambda$ from each side and taking a limit as $\lambda$ approaches
0, we obtain:

$\displaystyle\frac{\partial f_{n}}{\partial x_{i}}(0,\ldots,0)$
$\displaystyle=$ $\displaystyle\frac{\partial f_{n}}{\partial
x_{i}}(x_{1},\ldots,x_{n}).$

Thus all partial derivatives are constant. Since $f_{n}(0,\ldots,0)=0$ by
homogeneity, $f_{n}$ has the form:

$\displaystyle f_{n}(x_{1},\ldots,x_{n})$ $\displaystyle=$ $\displaystyle
a_{1}x_{1}+\ldots+a_{n}x_{n}.$

Symmetry now dictates that $a_{1}=\ldots=a_{n}$ and the fact that
$f_{n}(1,\ldots,1)=f_{n}(0,\ldots,0)+1=0+1=1$ accordingly implies that each
$a_{i}=\frac{1}{n}$.

The continuous differentiability assumption seems to be aimed primarily at
elimination of the median. So we reject it. Instead we offer as an axiom a
different property characteristic of the mean.

M-4) (Condensation) For $n>m$,

$\displaystyle f_{n}(x_{1},\ldots,x_{n})$ $\displaystyle=$ $\displaystyle
f_{n}(f_{m}(x_{1},\ldots,x_{m}),\ldots,f_{m}(x_{1},\ldots,x_{m}),x_{m+1},\dots,x_{n})$

for all $(x_{1},\ldots,x_{n})\in\mathcal{R}^{n}$.

This property asserts that if a subset of data is replaced by its “mean”, the
grand “mean” is not changed. This is the first property that proposes a
definite relationship between means of sets of different sizes. In view of the
symmetry axiom (M-3), the statement does not really restrict the order of the
subset, and as we shall see shortly the statement is only really needed in
special cases.

###### Proposition 2.3

Let $f_{n}:\mathcal{R}^{n}\rightarrow\mathcal{R}$ be a function satisfying
homogeneity, symmetry, translation invariance, and condensation, that is, M-1,
M-2, M-3, and M-4. Then

$\displaystyle f_{n}(x_{1},\ldots,x_{n})=\frac{x_{1}+\ldots+x_{n}}{n}$

for all $(x_{1},\ldots,x_{n})\in\mathcal{R}^{n}$.

Proof: In view of Proposition 2.1 we need only perform an inductive step
showing that the mean formula holds for $n\geq 3$ when it holds for $n-1$.
Consider

$\displaystyle f_{n}(x_{1},\ldots,x_{n})$ $\displaystyle=$ $\displaystyle
f_{n}(\frac{1}{n-1}(x_{1}+\ldots x_{n-1}),\ldots,\frac{1}{n-1}(x_{1}+\ldots
x_{n-1}),x_{n})$ $\displaystyle=$ $\displaystyle
f_{n}(0,\ldots,0,x_{n}-\frac{1}{n-1}(x_{1}+\ldots
x_{n-1}))+\frac{1}{n-1}(x_{1}+\ldots x_{n-1})$ $\displaystyle=$
$\displaystyle(x_{n}-\frac{1}{n-1}(x_{1}+\ldots
x_{n-1}))f_{n}(0,\ldots,0,1)+\frac{1}{n-1}(x_{1}+\ldots x_{n-1})$
$\displaystyle=$ $\displaystyle a_{1}x_{1}+...+a_{n}x_{n}.$

This shows that $f_{n}$ is a linear function of $x_{1},\ldots,x_{n}$. It now
follows from Proposition 2.2 that it is the mean.

The proof of Proposition 2.3 requires that M-4 holds in the case when $m=n-1$.
In fact we can get by with the assumption that M-4 holds when $m=2$. It is
easy to see that in this case M-4 also holds for $m=2^{k}$. Now set
$n=2^{k}+j$, where $0\leq j<2^{k}$. If $j=0$,
$f_{n}(x_{1},...,x_{n})=f_{n}(c,...,c)=cf_{n}(1,..,1)$ where
$c=(x_{1}+...+x_{n})/n$. If $j>0$, then
$f_{n}(x_{1},...,x_{n})=f_{n}(c,...,c,x_{m+1},...x_{n})$ where $m=2^{k}$ and
$c=(x_{1}+...+x_{m})/m$. We now replace $x_{1},...,x_{m}$ by
$x_{1}^{\prime},...,x_{m-j}^{\prime},0,...,0$ so that
$c=(x_{1}+...+x_{m})/m=(x_{1}^{\prime}+...+x_{m-j}^{\prime}+0+...+0)/m$. Using
the symmetry and homogeneity axioms, we obtain:
$f_{n}(x_{1},...,x_{n})=f_{n}((mc+x_{m+1}+...+x_{n})/m,...,(mc+x_{m+1}+...+x_{n})/m,0,...0)=((mc+x_{m+1}+...+x_{n})/m)f_{n}(1,...,1,0,...,0)=(x_{1}+...+x_{n})/m)f_{n}(1.,,,.1,0,...,0)$.
Thus we have established that $f_{n}$ is linear in $x_{1},...,x_{n}$. By
Proposition 2.3 $f_{n}$ is the ordinary mean.

A further note on axiomatics is that the translation invariance axiom can be
replaced by $f_{n}(1,...,1)=1$ if we also assume that
$f_{2}(x_{1},x_{2})=(x_{1}+x_{2})/2$.

To pass from the sample mean of a finite set to the usual general notion of
mean, we introduce a real-valued random variable $X$. We suppose that
associated with $X$ is a Borel probability measure $P_{X}$ taking each Borel
subset A of the real numbers to:

$\displaystyle P_{X}(A)=\mbox{ the probability that X belongs to the set A}.$

The mean of X, denoted by E(X) or $\mu_{X}$, is defined when $x$ is integrable
with respect to $P_{X}$ to be:

$\displaystyle E(X)=\int_{\mathcal{R}}xP_{X}(dx).$

One direction of the remarkable Strong Law of Large Numbers (see Pollard [12,
p. 78 and pp. 37-8]) states that if $\\{X_{n}\\}$ is a sequence of independent
random variables with common distribution $P_{X}$ and there exists a constant
$m$ such that

$\displaystyle\frac{X_{1}+\ldots+X_{n}}{n}\mbox{ converges almost surely to
}m$

as $n\rightarrow\infty$, then each $X_{n}$ has mean $m$. Here “almost surely”
means outside a set of measure zero in the countably infinite product space
induced by the measure $P_{X}$ (see [12, pp. 99-102]). More briefly, if sample
means of independent copies of $X$ settle down to something, then that
something is $E(X)$. This can be regarded as the motivation for the transition
from the sample mean to the mathematical mean $E(X)$. The general notion of
mean is derived from the finitary notion considered earlier.

The other direction of the Strong Law of Large Numbers asserts that if $E(X)$
exists, then the sample mean of $n$ identical independent copies of $X$
converges almost surely to $E(X)$ as $n$ tend to infinity. For a proof of both
directions of the Strong Law, see [12, pp. 95-102, p. 105]. For an alternate
proof due to N. Etemadi, see [12, pp. 106-7].

The transition here from finite samples to infinite populations distinguishes
the deductive method from the inductive method. While true science deals
comfortably with induction based on finite samples, the deductive method of
the Greeks (and Isaac Newton) relies on axioms whose relationship to reality
is only approximate and always contingent.

Indeed, in using the Strong Law of Large Number we are admittedly introducing
the full panoply of probability theory. It is possible, as noted in the
Introduction, to represent all of probability theory using the mean as the
primitive notion. Thus $P_{X}(A)$ can be defined as $E(\chi_{A}(X))$, the mean
of $\chi_{A}(X)$, where $\chi_{A}(X)$ is the random variable that equals 1
when $X$ is in $A$ and $0$ when $X$ is not in $A$. However, since in what
follows we plan to use probability theory in its conventional form (i.e.,
according to the axioms of Kolmogorov [6]), we see no reason to restate
measure-theoretic facts in terms of the mean as primitive. Indeed, a reason
not to do so is that the concept of independence, which is also fundamental in
probability, is awkward when expressed exclusively in terms of means.

##  3 Extending the Mean

When $x$ is not integrable with respect to $P_{X}$, the notion $E(X)$ above is
inapplicable and we must rely on other notions of mean. Richard Feynman was
famous for his integration tricks, and some of these are recorded in the book
of Mathews and Walker [11], based on lectures Feynman gave at Cornell.
Feynman’s tricks partly motivated our investigation.

Perhaps the most obvious generalization is the following:

$\displaystyle\mbox{Let
}L(c)=\lim_{M\rightarrow\infty}\int_{[c-M,c+M]}\,x\,P_{X}(dx)$

for a real number $c$.

By the Lebesgue Dominated Convergence Theorem this notion coincides with the
ordinary mean when $x$ is integrable with respect to $P_{X}$. Kolmogorov [6,
p. 40], in his great foundational work, noted this option in the case when
$c=0$ and observed that it does not require integrability of $|x|$. Indeed if
$X$ is a random variable obeying the Cauchy distribution
$f(x)=1/\pi(1+x^{2})$, then $X$ satisfies $L(c)\equiv 0$ for any choice of
$c$.

We mention two related notions of mean:

  1. L-1)

$\lim_{M\rightarrow\infty}\int_{[a-M,b+M]}\,xP_{X}(dx)\mbox{, and}$

  2. L-2)

$\lim_{\min{\\{M,K\\}}\rightarrow\infty}\int_{[a-M,b+K]}\,xP_{X}(dx),$

where $a\leq b$.

It is easily seen that L-1 coincides with $L((a+b)/2)$ since

$[a-M,b+M]=[\frac{a+b}{2}-(M+\frac{b-a}{2}),\frac{a+b}{2}+(M+\frac{b-a}{2})].$

As for L-2, we have the following result.

###### Proposition 3.1

Let X be a random variable with probability measure $P_{X}$. Then for some $a$
and $b$ with $a\leq b$

$\lim_{\min{\\{K,M\\}}\rightarrow\infty}\int_{[a-M,b+K]}\,xP_{X}(dx)$

exists if and only if $x$ is integrable with respect to $P_{X}$.

Proof: If $x$ is integrable on $\mathcal{R}$, the limit exists and equals the
mean of $x$ by Lebesgue’s Dominated Convergence Theorem. Conversely, if the
limit exists, then

$0\leq\int_{(b+K,b+K^{\prime}]}x\,P_{X}(dx)<\epsilon$

for $K<K^{\prime}$, both sufficiently large, and any given $\epsilon$.
Likewise

$-\epsilon<\int_{[a-M^{\prime},a-M)}x\,P_{X}(dx)\leq 0$

for $M<M^{\prime}$, both sufficiently large. Fatou’s Lemma or Levi’s Theorem
[4, p. 172] thus implies that

$0\leq\int_{(b+k,\infty)}x\,P_{X}(dx)\leq\epsilon,\hskip
14.45377pt-\epsilon\leq\int_{(-\infty,a-M)}x\,P_{X}(dx)\leq 0$

and thus $x$ is integrable on $[0,\infty)$ as well as $(\infty,0]$ and so is
integrable on $\mathcal{R}=(-\infty,\infty)$.

So, when L-2 exists, it coincides with E(X).

We now return to the study of $L(c)$. We shall allow $-\infty\leq
L(c)\leq\infty$. This gives us a bit more flexibility in characterizing what
can happen.

###### Lemma 3.1

Let X be a random variable with probability measure $P_{X}$, and let $c_{1}$
and $c_{2}$ be real numbers with $c_{1}<c_{2}$. Then there are three
possibilities:

i) If $L(c_{1})$ exists in $[-\infty,\infty]$, then

$L(c_{1})\leq\liminf_{M\rightarrow\infty}\int_{[c_{2}-M,c_{2}+M]}\,x\,P_{X}(dx);$

ii) If $L(c_{2})$ exists in $[-\infty,\infty]$, then

$L(c_{2})\geq\limsup_{M\rightarrow\infty}\int_{[c_{1}-M,c_{1}+M]}\,x\,P_{X}(dx);$

iii) If $L(c_{1})$ and $L(c_{2})$ both exist in $[-\infty,\infty]$ , then
$L(c_{1})=L(c_{2})$.

Proof: Suppose $c_{1}<c_{2}$. Then

$\displaystyle\int_{[c_{2}-M,c_{2}+M]}\,xP_{X}(dx)=\int_{[c_{1}-M,c_{1}+M]}\,xP_{X}(dx)$
$\displaystyle+\int_{(c_{1}+M,c_{2}+M]}\,xP_{X}(dx)-\int_{[c_{1}-M,c_{2}-M)}\,xP_{X}(dx).$

The second and third terms on the right are both non-negative and accordingly
i) and ii) follow. In the case of iii), note that i) and ii) imply that if
both $L(c_{1})$ and $L(c_{2})$ exist, then $L(c_{1})\leq L(c_{2})$.

If $L(c_{2})-L(c_{1})>0$, then there is a positive constant $K$ (for example,
any positive number $<L(c_{2})-L(c_{1})$) such that for $M$ sufficiently
large:

$\displaystyle
K<\int_{(c_{1}+M,c_{2}+M]}\,xP_{X}(dx)-\int_{[c_{1}-M,c_{2}-M)}\,xP_{X}(dx)$
$\displaystyle\leq$
$\displaystyle(c_{2}+M)P_{X}((c_{1}+M,c_{2}+M])+(M-c_{1})P_{X}([c_{1}-M,c_{2}-M))$
$\displaystyle\leq$
$\displaystyle(M+d)(P_{X}((c_{1}+M,c_{2}+M]\cup[c_{1}-M,c_{2}-M))$

where $d=\max{\\{|c_{2}|,|c_{1}|\\}}$. Thus

$\frac{K}{M+d}<P_{X}((c_{1}+M,c_{2}+M]\cup[c_{1}-M,c_{2}-M)).$

Now replace $M$ by $M_{j}=M+j(c_{2}-c_{1})$ for each integer $j\geq 0$ to get:

$\frac{K}{M_{j}+d}<P_{X}((c_{1}+M_{j},c_{2}+M_{j}]\cup[c_{1}-M_{j},c_{2}-M_{j})).$

Summing over these inequalities and noting that $c_{2}+M_{j}=c_{1}+M_{j+1}$
and $c_{1}-M_{j}=c_{2}-M_{j+1}$, we obtain:

$\infty=\sum_{j=0}^{\infty}\frac{K}{M+d+j(c_{2}-c_{1})}\leq
P_{X}((c_{1}+M,\infty)\cup(-\infty,c_{2}-M))\leq 1$

for a contradiction. Thus, this case is eliminated. So $L(c_{1})=L(c_{2})$.

###### Theorem 3.1

Let X be a random variable with probability measure $P_{X}$. Then exactly one
of the following possibilities holds:

i) $L(c)$ does not exist in $[-\infty,\infty]$ for any real number $c$;

ii) $L(c)$ exists in $(-\infty,\infty)$ for exactly one real number $c$;

iii) $L(c)$ exists in $[-\infty,\infty]$ for all real numbers $c$ and is
independent of $c$;

iv) there is a number $c_{0}$ such that $L(c)=\infty$ for $c>c_{0}$ and $L(c)$
does not exist for $c<c_{0}$; or

v) there is a number $c_{0}$ such that $L(c)=-\infty$ for $c<c_{0}$ and $L(c)$
does not exist for $c>c_{0}$.

Proof: By Lemma 3.1 it suffices to show what happens when $L(c_{2})=L(c_{1})$
is finite. In this case the last two terms in the equation at the beginning of
the proof of Lemma 3.1 each tend to $0$ as $M$ tends to infinity. By a change
of variable, we obtain for the positive number $c=c_{2}-c_{1}$.

$\lim_{M\rightarrow\infty}\int_{(M,c+M]}xP_{X}(dx)=\lim_{M\rightarrow\infty}\int_{[-M-c,-M)}xP_{X}(dx)=0;$

Assume $0<d<c$ and $M\geq 0$. Then

$0\leq\int_{(M,d+M]}xP_{X}(dx)\leq\int_{(M,c+M]}xP_{X}(dx)$

and

$\int_{[-M-c,-M)}xP_{X}(dx)\leq\int_{[-M-d,-M)}xP_{X}(dx)\leq 0$

Thus if ii) holds for $c$, it holds for $d$. On the other hand, if ii) holds
for $c$ it also holds for $nc$ where $n$ is any fixed positive integer since

$\int_{(M,nc+M]}xP_{X}(dx)=\sum_{j=1}^{n}\int_{((j-1)c+M,jc+M]}xP_{X}(dx)$

and

$\int_{[-M-nc,-M)}xP_{X}(dx)=\sum_{j=1}^{n}\int_{[-M-(n+1-j)c,-M-(n-j)c)}xP_{X}(dx),$

and if the $M^{\prime}s$ are chosen far enough out so that the individual
integrals are closer to zero than $\epsilon/n$, the sum integral is within
$\epsilon$ of $0$. Finally since any positive real number $d$ is smaller than
$nc$ for some positive integer $n$, all cases are covered. Accordingly, ii)
implies iii).

The argument for i) implies ii) can now be used to show that for any two real
numbers $c_{1}$ and $c_{2}$, if either $L(c_{1})$ or $L(c_{2})$ exists in
$[-\infty,\infty]$, then the other exists and equals it since the
approximating integrals differ by two integrals on intervals of length
$|c_{1}-c_{2}|$ that tend to zero as $M$ tends to infinity. Thus iii) implies
iv).

We give some examples to illustrate that each of the possibilities enumerated
in Theorem 3.1 can occur.

Consider a random variable $X$ whose probability measure is of the form

$P_{X}(A)=\sum_{n=1}^{\infty}(\frac{1}{2^{2n}}\delta_{2^{2n}}(A)+\frac{1}{2^{2n-1}}\delta_{-2^{2n-1}}(A))$

where $\delta_{z}$ is the (Dirac) probability measure whose value is $1$ on
any Borel subset $A$ of $\mathcal{R}$ that contains the real number $z$ and
whose value is zero otherwise. The sum of the nonzero values is one, so this
obviously defines a probability measure. However the integral of $x$ over the
interval $[c-M,c+M]$ is the difference between the size of the first set and
the size of the second set below:

$\displaystyle\,\mbox{ The size of the set }\\{n\mbox{ : }1\leq
n,2^{2n}\leq(c+M)\\}$ $\displaystyle=$
$\displaystyle\lfloor{\frac{\log{(c+M)}}{2\log{2}}}\rfloor$
$\displaystyle\mbox{ The size of the set }\\{n\mbox{ : }1\leq
n,2^{2n-1}\leq(M-c)\\}$ $\displaystyle=$
$\displaystyle\lfloor{\frac{\log{(M-c)}}{2\log{2}}+\frac{1}{2}}\rfloor$

where $\lfloor\,\,\rfloor$ is the floor function. For fixed $c$ and
sufficiently large $M$ the difference of the above quantities can assume the
values $0$ and $-1$ and the integral does not settle down to either one. This
is an instance of Theorem 3.1, part i).

A random variable $X$ can also be defined with probability measure of the form

$P_{X}(A)=\sum_{n=1}^{\infty}\frac{1}{2^{n+1}}(\delta_{2^{n}}(A)+\delta_{(-2^{n})}(A)).$

So $P_{X}$ is concentrated at the points $\pm 2^{n}$ and assigns probability
$1/(2^{n+1})$ to such points. For this measure, $L(0)$ equals $0$ by symmetry.
However, $L(c)$ does not exist for other choices of $c$. If $c$ is positive,
the integral of $x$ over the closed interval $[c-M,c+M]$ reduces to its
integral over the open interval $(M-c,M+c]$ and this integral oscillates
between $0$ and $\frac{1}{2}$ for large $M$ depending on whether $2^{n}$ is in
the interval $(M-c,M+c]$ or not. Similar behavior occurs when $c<0$. This
example is an instance of Theorem 3.1, part ii).

Now consider a random variable $X$ having a probability density (with respect
to Lebesgue measure on the real line) of the form

$f(x)=\left\\{\begin{array}[]{ll}\frac{1}{1+Cx^{a}}\mbox{ if $x\geq 0$}&\\\
\frac{1}{1+D|x|^{b}}\mbox{ if $x<0$}&\end{array}\right.$

where $a$ and $b$ are numbers in $(1,2)$ and $C$ and $D$ are suitable positive
constants that guarantee that the density integrates to $1$. Notice that this
random variable satisfies iii) of Proposition 3.1. It is easy to see that
$L(c)\equiv\infty$ for all $c$ or $-\infty$ for all $c$ according as $b>a$ or
$a>b$. The example illustrates part iii) of Theorem 3.1 (as does the Cauchy
distribution with $L(c)\equiv 0$) .

The probability measure

$P_{X}(A)=\sum_{n=1}^{\infty}(\frac{2^{n-1}}{3^{n+1}}\delta_{3^{n}}(A)+\frac{2^{n-2}}{3^{n+1}}\delta_{-3^{n}}(A))$

illustrates part iv) of Theorem 3.1. If $c\geq 0$, the integral of $x$ over
$[c-M,c+M]$ is given by:

$\sum_{\\{n\mbox{ : }1\leq n,3^{n}\leq
c+M\\}}\frac{2^{n-1}}{3}-\sum_{\\{n\mbox{ : }1\leq n,3^{n}\leq
M-c\\}}\frac{2^{n-2}}{3},$

and this expression has the value $(2^{n_{0}-1}-2^{-1})/3$ or
$(2^{n_{0}-1}+2^{n_{0}-2}-2^{-1})/3$ where $n_{0}\approx(\log{c+M})/\log{3}$
for large $M$. Since $M$ and $n_{0}$ tend to infinity together, it follows
that $L(c)\equiv\infty$ for $c\geq 0$.

On the other hand, if $c=-d$ where $d>0$, the integral of $x$ over
$[c-M,c+M]=[-M-d,M-d]$ is given by:

$\sum_{\\{n\mbox{ : }1\leq n,3^{n}\leq
M-d\\}}\frac{2^{n-1}}{3}-\sum_{\\{n\mbox{ : }1\leq n,3^{n}\leq
M+d\\}}\frac{2^{n-2}}{3},$

and this reduces to $(2^{n_{0}-1}-2^{-1})/3$ or to $(-1)/6$ for large $M$
where $n_{0}\approx(\log{M-d})/\log{3}$ depending on whether a positive
integer lies in the interval $(\log{M-d}/\log{3},\log{M+d}/\log{3}]$ or not.
Thus, $L(c)$ does not exist for $c<0$.

A final example (also for part iv) of Theorem 3.1 is the case where the
probability measure is given by:

$P_{X}(A)=K\sum_{n=1}^{\infty}(\frac{2^{n}}{3^{n}+(1/n)}\delta_{3^{n}+(1/n)}(A)+\frac{2^{n-1}}{3^{n}}\delta_{-3^{n}}(A)).$

Here $K$ is a suitably chosen positive normalizer, which is easily seen to be
smaller than $1/3$. For $c>0$, the integral of $x$ over $[c-M,c+M]$ is

$K\sum_{\\{n\mbox{ : }1\leq n,(3^{n}+1/n)\leq M+c\\}}2^{n}-K\sum_{\\{n\mbox{ :
}1\leq n,3^{n}\leq M-c\\}}2^{n-1},$

and this reduces to $K(2^{n_{0}}-1)$ for $M$ sufficiently large where $n_{0}$
is the largest integer such that $3^{n_{0}}\leq M-c$. Since $n_{0}$ and $M$
tend to infinity together, $L(c)\equiv\infty$ for all $c>0$.

When $c=0$, the integral of $x$ over $[-M,M]$ reduces to $K(2^{n_{0}}-1)$ or
to $-K$ where $n_{0}$ is the largest integer such that
$3^{n_{0}}+(1/n_{0})\leq M$ and the first or second reduction occurs according
as $M<3^{n_{0}+1}$ or not. Thus $L(0)$ does not exist. By Theorem 3.1 $L(c)$
does not exist for $c<0$.

Other cases arising in Theorem 3.1, such as part v), are obtained by modifying
the examples above, e.g., replacing $X$ by $-X$ or by $X+a$.

##  4 Weak Means and Multipliers

One of the implications of Theorem 3.1 is that if $L(c)$ exists for more than
one choice of $c$ and is finite in some case then it is finite for all $c$ and
is independent of $c$. The case of the Cauchy distribution shows that this can
happen without the ordinary mean existing. Accordingly for a random variable
$X$, we define the doubly weak mean of $X$, denoted by $E_{ww}(X)$, to be the
common value of $L(c)$ for all $c$ when this common value exists and is in
$(-\infty,\infty)$.

We also introduce an intermediate notion due to Kolmogorov between the
ordinary mean and the doubly weak that motivates our terminology. The weak
mean of $X$, denoted by $E_{w}(X)$, is defined as follows: $E_{w}(X)$ is the
quantity $L(0)$ provided the latter exists in $(-\infty,\infty)$ and provided
$\lim_{n\rightarrow\infty}nP_{X}(|X|>n)=0$.

The following proposition is due to Kolmogorov. It indicates that existence of
the weak mean coincides precisely with the existence of a number for which the
Weak Law of Large Numbers holds.

######  Proposition 4.1 (Kolmogorov, 1928)

Let $X$ be a random variable. Suppose that $\\{X_{1},\cdots,X_{n},\cdots\\}$
are independent identically distributed copies of $X$ with $P_{n}$ the n-fold
product distribution. Then there is a real number $m$ such that for each
$\epsilon>0$

$\lim_{n\rightarrow\infty}{P_{n}(|\frac{X_{1}+...+X_{n}}{n}-m|>\epsilon)}=0$

if and only if $X$ has weak mean $E_{w}(X)=m$.

Proof: See [6, p. 65], [7], and [8, Theorems XII and XIII].

###### Corollary 4.1

Let $X$ be a random variable.

i) If $X$ has a mean, then $X$ has a weak mean and $E(X)=E_{w}(X)$;

and

ii) if $X$ has a weak mean, then $X$ has a doubly weak mean and
$E_{w}(X)=E_{ww}(X)$.

Proof: In case $X$ has a mean, then the identity function $x\mapsto x$ is
integrable with respect to the probability measure $P_{X}$ on the real line.
In particular the tail integrals

$\int_{[n,\infty)}xP_{X}(dx)\mbox{ and }\int_{(-\infty,-n]}xP_{X}(dx)$

tend to zero as $n$ tends to infinity. Since the absolute values of these
integrals are larger respectively than $nP_{X}(X\geq n)$ and
$nP_{X}(X\leq-n)$, it follows that $\lim_{n\rightarrow\infty}nP_{X}(|X|>n)=0$.
Likewise by Lebesgue’s Dominated Convergence Theorem, $L(0)=E(X)$, Suppose $X$
has a weak mean. Then if $c_{1}<c_{2}$ and $\epsilon>0$ and a sufficiently
large $M$ are given,

$\displaystyle 0\leq\int_{(c_{1}+M,c_{2}+M]}\,xP_{X}(dx)$ $\displaystyle\leq$
$\displaystyle(c_{2}+M)P_{X}((c_{1}+M,c_{2}+M])$ $\displaystyle\leq$
$\displaystyle(c_{2}-c_{1}+c_{1}+M)P_{X}(|X|>c_{1}+M)$ $\displaystyle\leq$
$\displaystyle(c_{2}-c_{1}+c_{1}+M)P_{X}(|X|\geq n)$ $\displaystyle\leq$
$\displaystyle(\frac{c_{2}-c_{1}}{n}+\frac{c_{1}+M}{n})\epsilon$

where $n=\lfloor{c_{1}+M}\rfloor$. For $M$ sufficiently large, the right side
is as close to $\epsilon$ as we like. Thus

$\lim_{M\rightarrow\infty}\int_{(c_{1}+M,c_{2}+M]}\,xP_{X}(dx)=0.$

Similarly,

$\lim_{M\rightarrow\infty}\int_{[c_{1}-M,c_{2}-m)}\,xP_{X}(dx)=0.$

Accordingly from the first equation in the proof of Lemma 3.1, it follows that
when one of $L(c_{2})$ or $L(c_{1})$ exists and is finite, the other exists
and is equal to it. Since $L(0)=m$, it follows that $L(c)$ exists for all $c$,
$L(c)\equiv m$ and $m$ is the doubly weak mean of $X$.

Kolmogorov in [6, p. 66] gives an example where the Weak Law holds but the
Strong Law does not. Cauchy random variables have $L(c)$ existing for all $c$,
independent of $c$, but violate the Weak Law by not decaying rapidly enough at
infinity. Thus the mean, weak mean, and doubly weak mean are strictly distinct
notions.

We make one more observation on generalizations of the mean, based on using
multipliers to attempt to finitize the mean. These multipliers are a type of
“mollifier.” Usually mollifiers are used to aid approximation of the delta
function and to smooth functions, but another use is to regularize behavior at
$\pm\infty$. The idea is to introduce a function $\phi_{\lambda}(x)$ that
depends on a parameter $\lambda$ so that $x\mapsto\phi_{\lambda}(x)x$ is
integrable with respect to $P_{X}$ for $\lambda\neq\lambda_{0}$ and
$\phi_{\lambda}(x)\rightarrow 1$ for a. e. x as
$\lambda\rightarrow\lambda_{0}$. In the case of L(c), the multiplier can be
taken to be

$\phi_{\lambda}(x)=\chi_{[c-1/\lambda,c+1/\lambda]}(x)$

where $\chi_{A}$ is the characteristic function of the set $A$ and
$\lambda=1/M$.

Multipliers, and indeed other straight-forward generalizations of the mean
including the weak and doubly weak mean, are useful only when the following
equations hold:

$\displaystyle\int_{[0,\infty)}xP_{X}(dx)$ $\displaystyle=$
$\displaystyle\infty$ $\displaystyle\int_{(-\infty,0]}xP_{X}(dx)$
$\displaystyle=$ $\displaystyle-\infty.$

If neither of these equations holds, x is integrable and the mean is well-
defined. If only the first equation holds, the mean is $+\infty$, and if only
the second equation holds, the mean is $-\infty$. If both equations hold, then
there is some room for maneuver. $L(c)$ cannot exist finitely unless the
infinities on each end are of the same order. If for example $P_{X}$ is given
by a density function $f$ with respect to Lebesgue measure such that $f(x)$
decays as $1/x^{2}$ as $x\rightarrow\infty$ and decays as $1/|x|^{3/2}$ as
$x\rightarrow-\infty$, then $L(c)\equiv-\infty$ for all $c$.

Multipliers offer possibilities for extending the notion of the mean. They can
be of use in such activities as renormalization where the aim is to
reinterpret integrals to make them finite. In our case we set:

$E_{mult}(X)=\lim_{\lambda\rightarrow\lambda_{0}}{E(\phi_{\lambda}(X)X)}$

provided this limit exists. This method is used to “evaluate” the integrals of
$\sin{bx}$ and $\sin{x}/x$ on $[0,\infty)$ in [11, p. 60 and p. 91].)

However, there are dangers that the following example illustrates.

Define a function $\phi_{\lambda,c}$ for $\lambda>0$ and $c$ in $\mathcal{R}$
by:

$\phi_{\lambda,c}(x)=\left\\{\begin{array}[]{ll}e^{-\lambda x}&\mbox{if
$x>0$}\\\ e^{\lambda x}(1+\pi c\lambda x)&\mbox{if $x<0$.}\end{array}\right.$

Here $c$ is an arbitrary constant. Evidently $\phi_{\lambda,c}$ is a well-
behaved function, integrable and dying off at $\pm\infty$. Also
$\\{\phi_{\lambda,c}\\}$ converges pointwise to the constant function one as
$\lambda$ tend to $0^{+}$ with fixed $c$.

Suppose we use this family of functions as a multiplier to determine a mean
for a variable obeying the Cauchy distribution. Let $m(\lambda,c)$ be defined
by:

$m(\lambda,c)=\int_{-\infty}^{\infty}\phi_{\lambda,c}(x)\frac{x}{\pi(1+x^{2})}\,dx=\int_{-\infty}^{0}\frac{c\lambda
e^{\lambda x}x^{2}}{1+x^{2}}\,dx=cm(\lambda,1).$

Now

$\displaystyle 1=e^{\lambda x}|_{-\infty}^{0}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{0}\lambda e^{\lambda x}\,dx$
$\displaystyle\geq\int_{-\infty}^{0}\frac{\lambda e^{\lambda
x}x^{2}}{1+x^{2}}\,dx$ $\displaystyle=$ $\displaystyle
m(\lambda,1)=\int_{0}^{\infty}\frac{\lambda e^{-\lambda x}x^{2}}{1+x^{2}}\,dx$
$\displaystyle\geq\int_{K}^{\infty}\frac{\lambda e^{-\lambda
x}x^{2}}{1+x^{2}}\,dx$ $\displaystyle\geq$
$\displaystyle\frac{K^{2}e^{-\lambda K}}{1+K^{2}}$

for any positive real number $K$. Thus

$1\geq\limsup_{\lambda\rightarrow
0+}m(\lambda,1)\geq\liminf_{\lambda\rightarrow
0+}m(\lambda,1)\geq\frac{K^{2}}{1+K^{2}}.$

Letting $K$ tend to infinity, we find that $\lim_{\lambda\rightarrow
0+}m(\lambda,1)=1$.

Hence the multiplier-induced mean of the standard Cauchy distribution is:

$\displaystyle E_{mult}(X)=\lim_{\lambda\rightarrow
0+}\int_{-\infty}^{\infty}\phi_{\lambda,c}(x)\frac{x}{\pi(1+x^{2})}\,dx$
$\displaystyle=$ $\displaystyle\lim_{\lambda\rightarrow 0+}m(\lambda,c)$
$\displaystyle=\lim_{\lambda\rightarrow 0+}cm(\lambda,1)$ $\displaystyle=$
$\displaystyle c\lim_{\lambda\rightarrow 0+}m(\lambda,1)=c.$

However, $c$ was arbitrary depending on the choice of the multiplier!

Although some may consider the Cauchy distribution anomalous, we remind the
reader that its legitimacy and importance stem in part from the fact that it
is the quotient of two independent standard normal random variables. It has
application in physics under the name of the Lorentz distribution. Indeed
long-tailed and counter-intuitive distributions are increasingly important in
recent times (see Gumble [3] or Taleb [13]) in financial mathematics, the
study of natural and man-made disasters, and computer network analysis.
Extending the notion of mean to such distributions, and investigating the
limits of the notion of mean in such settings, are among the ways of moving
beyond the normal regime.

##  5 State Space Theory

State Space Theory or System Theory is widely used to provide a mathematical
description of physical systems including those of classical mechanics as well
as other systems such as biological and social systems. The state of the
system at any time is taken to be an element of a set $S$ called state space.
The evolution of the state is given by a function $T_{t}:S\rightarrow S$
taking the state $s$ at time $0$ to the state $T_{t}(s)$ at time t. A (real-
valued) _observable_ is any function $f:S\rightarrow R$ which assigns to each
state $s$ a number $f(s)$ (see Mackey[10]). All observables may be determined
from the state, and indeed the state can be viewed as a maximal independent
set of observables that characterize the system at a given time. The dynamic
evolution of the state is deterministic and time may be taken to be either
discrete or continuous. Evolution of an observable $f$ can be expressed by
$t\mapsto f\circ T_{t}(s)$, i.e., the value of the observable at time t is
obtained by applying the observable function to the state at time t.

A familiar example of the state space approach is Hamiltonian mechanics. The
state space in this case is phase space, and a state is a 2n-tuple
$(q,p)=(q_{1},...,q_{n},p_{1},...,p_{n})$ consisting of position coordinates
$q_{i}$ and momentum coordinates $p_{i}$. The evolution is
$T_{t}(q,p)=(q(t),p(t))$, where the latter is the solution to Hamilton’s
equations with initial data $(q(0),p(0))=(q,p)$:

$\displaystyle\frac{dq_{i}}{dt}$ $\displaystyle=$ $\displaystyle\frac{\partial
H}{\partial p_{i}}$ $\displaystyle\frac{dp_{i}}{dt}$ $\displaystyle=$
$\displaystyle-\frac{\partial H}{\partial q_{i}}$

for $i=1,2,...,n$. Here $H(q,p)$ is the Hamiltonian function of the system,
which is assumed to be a continuously differentiable function on state space
representing the total energy of the system. The function $H$ is an example of
an observable, as are the position and momentum coordinates, angular momenta
$q_{i}p_{j}-q_{j}p_{i}$, et cetera. A differentiable observable $f$ evolves
according to the equation:

$\frac{df}{dt}=\sum_{i=1}^{n}(\frac{\partial f}{\partial q_{i}}\frac{\partial
H}{\partial p_{i}}-\frac{\partial f}{\partial p_{i}}\frac{\partial H}{\partial
q_{i}}),$

the right-hand side being the definition of the Poisson bracket $[f,H]$, under
which operation $C^{\infty}$ observables form a Lie algebra.

Given a deterministic state space it is natural to pass to a statistical
setting as follows. We replace the old states $s$ by new states that are
(Borel) probability measures $P$ on the state space $S$. The old observables
$f$ on the original state space are replaced by new observables that are the
means of the old observables with respect to the probability measure $P$. Thus
for any original state space observable $f$, the map

$P\mapsto E_{P}(f)$

defines an observable on the set of probability measures. If $f$ is bounded,
this observable is defined for all probability measures. If not, it is defined
for those measures with respect to which $f$ is integrable.

If the only observables allowed were obtained in this manner, this would
appear to be a severe limitation. However, the variance of f and all moments
of f can themselves be regarded as means of original observables. Indeed, even
the probability distribution for $f$ can be regarded as a mean. This is
because on a Borel subset $A$ of the reals, the probability that $f$ takes a
value in $A$ is given by $E_{P}(\chi_{A}\circ f)$, where $\chi_{A}$ is the
characteristic function of the set $A$.

The evolution of the probabilistic state can be induced by an underlying
deterministic evolution. The probability measure at time $t$, $P_{t}$, is
given by $P_{t}(A)=P(T_{-t}(A))$ where $A$ is any (Borel) subset of $S$. This
permits us to talk about the evolution of observables since the mapping
$t\mapsto E_{P_{t}}(f)$ describes such an evolution. In the Hamiltonian
formalism phase space has a natural 2n-dimensional Lebesgue measure $\lambda$
called Liouville measure with infinitesimal volume element
$dq_{1}...dq_{n}dp_{1}...dp_{n}$, and $\lambda(T_{t}(A))=\lambda(A)$ for all
Borel subsets $A$ of $S$ and all times t. Dynamics in phase space can be
thought of as a fluid flow that permits change of shape but no change in
volume. The probability state $P$ can often be taken to be the integral of a
probability density function $\rho(q,p)$ with respect to $\lambda$. At the
other extreme $P$ can be taken to be a delta function
$\delta(q-q_{0})\delta(p-p_{0})$, which reduces to the deterministic theory
with state $s=(q_{0},p_{0})$. The probabilistic setting also permits us to
abandon the deterministic evolution $\\{T_{t}\\}$ and work with a stochastic
evolution exclusively, e.g., one of Markov type.

A use of means in state space theory that we have not touched on here relates
to ergodic theory, in which time averages of observables over trajectories are
compared with averages over state space regions using a suitable normalized
volume measure.

The essential point is that means provide the transition from classical
observables for deterministic systems to statistical observables for
stochastic systems.

##  6 Entropy

A subtlety occurs in statistical mechanics that is not present in ordinary
probability theory. An observable is commonly defined as a real-valued
function of the state, and in statistical mechanics the state is a probability
measure $P$ on state space. Thus any real-valued function of $P$ can be taken
to be an observable, e. g., $P\mapsto P(B)$ is an observable where $B$ is any
fixed Borel set in the state space $S$. This observable is an expected value
since $P(B)=E_{P}(\chi_{B})$. However, not all observables arise as expected
values of original observables. The most familiar example of such an
observable is the entropy function, which can be interpreted as an expected
value (mean) but is not a conventional mean.

To avoid certain difficulties associated with the continuous case we will
confine our attention to the case where the underlying state space is a finite
set. Let $S$ be a finite state consisting of n states. A classical observable
is a function $f:S\rightarrow(\infty,\infty)$. A discrete classical evolution
might be a function $T:S\rightarrow S$ such that if $i$ is the state at a
given time then $T(i)$ is the state one time unit later. (A continuum of times
presents a problem for deterministic evolution in a finite state space,
although that problem does not arise in the probabilistic setting.)

When we pass to a statistical notion of state, we arrive at a probability
vector $p=(p_{1},...,p_{n})$ where $p_{i}$ is the probability that the system
is in the deterministic state $i$. We can now form expected values of
classical observables $f$, i.e.,

$E(f)=\sum_{i}p_{i}f(i)$

as noted before. We can also form such expressions as the entropy:

$H(p)=\sum_{i}p_{i}\log(\frac{1}{p_{i}}).$

Superficially the entropy appears to be another mean value, the mean value of
the “uncertainty” $\log(1/p_{i})$, also called the “surprise value” (The log
here is usually taken with base $2$.) Thus the entropy of a probability state
is the mean uncertainty of the state. This is not the mean of a classical
observable since the function $i\mapsto\log(1/p_{i})$ is not a classical
observable. Classical observables should exist and be measurable prior to
assignment of probabilities, but it makes no sense to consider the uncertainty
function until probabilities have been introduced. The dependency of the
uncertainty function on $i$ is not intrinsic and is only determined through
the postulated probability state $p_{i}$.

It happens that entropy has another relationship to means of considerable
importance, namely through the _Maximum Entropy Principle (MEP)_ , also known
as _Jaynes’ Principle_. In the absence of an evolutionary law $T$ and an
initial assignment, we are faced with the problem of determining the
probability state $p$, i. e., an assignment of probabilities to the
deterministic states $i$. The MEP [5, p. 370] asserts that:

> The probability state $p$ maximizing entropy subject to the given values
> $\alpha_{1},...\alpha_{k}$ for the means of known classical observables
> $g_{1},...,g_{k}$ provides predictions ”most strongly indicated by our
> present information.”

Using the calculus of variations, we can in general determine a unique
distribution among those that satisfy the constraints

$\sum_{i}p_{i}g_{j}(i)=\alpha_{j}$

for $j=1,...,k$ and maximizing $H(p)$, namely, the one with the probability
assignment

$p_{i}=C\exp\\{-(\sum_{j}\beta_{j}g_{j}(i))\\}$

for $i=1,...,n$ where $\beta_{1},\dots,\beta_{k}$ are constants determined
from the $\alpha_{i}\mbox{'s}$, and $C$ is a positive normalizing constant
chosen so that the sum of the $p_{i}$’s is $1$.

The interpretation of this result takes two forms (at least). Suppose the
states are those of an individual particle in a gas of $N$ particles. Then the
quantities $\alpha_{1},...\alpha_{k}$ represent measured values of the total
value of $g_{1},...,g_{k}$ over the entire gas divided by $N$. The
probabilities $p_{i}$, derived from the MEP, are the probabilities that a
particle picked at random from among the $N$ particles is in the i-th state.
They may also be regarded as the fraction of particles that are in the i-th
state. We may not care about individual particles but we do care about these
fractions, which can be taken to define the macroscopic state of the gas
(volume, pressure, temperature, and the like). This is the _ensemble_
viewpoint of Gibbs. Yet another perspective is to regard what we usually
observe as a small perturbation about values induced by means.

##  7 Quantum Issues

The mean plays a pivotal role in quantum theory, even if this role has not
been examined closely in most treatments of quantum theory. In quantum
mechanics the state of a physical system is described by a wave function
$\psi$ that is an element of a Hilbert space $\mathcal{H}$. (Strictly speaking
$\psi$ is not a function but an equivalence class of functions, and in
addition each state is associated with a ray in Hilbert space.) Each physical
observable that takes on real-number values (e.g., a position coordinate, a
momentum coordinate, the energy, a spin component) is associate with a self-
adjoint operator $A$ in $\mathcal{H}$. For simplicity each observable is
denoted by the same symbol “A” as the associated operator. Any self-adjoint
operator A has in turn an associated projection-valued measure $P_{A}$ (see,
for example, [10]) that assigns to each Borel set $S$ in $\mathcal{R}$ an
orthogonal projection $P_{A}(S)$ in the Hilbert space:

$S\longmapsto P_{A}(S)$

in such a way that $A$ is an integral combination of these orthogonal
projections, represented symbolically by:

$A=\int_{\mathcal{R}}x\,P_{A}(dx),$

or by:

$A(\psi)=\int_{\mathcal{R}}x\,P_{A}(dx)(\psi),$

where $\psi$ is in the domain of $A$. If a measurement is made, the
probability that the value of A is in the set $S$ when the system state is
$\psi$ defined to be:

$\left\langle P_{A}(S)(\psi),\psi\right\rangle=||P_{A}(S)(\psi)||^{2}$

where $<\,,\,>$ is the inner product on $\mathcal{H}$, linear in the first
variable and conjugate-linear in the second variable, and $||\,||$ is the norm
on $\mathcal{H}$.

Quantum Mechanics is thus a statistical theory based on a family of
probability measures defined by:

$S\longmapsto||P_{A}(S)(\psi)||^{2}.$

These are the Borel probability measures associated with observables $A$ when
the system state is $\psi$. One consequence of this is that the set of
possible values of $A$ is the spectrum of the operator $A$, and another is
that the mean of $A$, when the state is $\psi$, is given by:

$\left\langle A(\psi),\psi\right\rangle=\int_{\mathcal{R}}x\,\left\langle
P_{A}(dx)(\psi),\psi\right\rangle=\int_{\mathcal{R}}x||P_{A}(dx)(\psi)||^{2}.$

In particular the quantity $\left\langle
P_{A}(S)(\psi),\psi\right\rangle=||P_{A}(S)(\psi)||^{2}$ can be interpreted as
the mean of the observable $P_{A}(S)$ when the state is $\psi$. The observable
$P_{A}(S)$ is an orthogonal projection, taking the value $1$ when the value of
$A$ is in is $S$ and the value $0$ when the value of $A$ is not in $S$. Thus
$||P_{A}(S)(\psi)||^{2}$ also represents the probability that $A$ is in $S$
when the state is $\psi$. This is a reminder that all probabilities are means.

The mean, $\left\langle A(\psi),\psi\right\rangle$, is the integral over the
real line of the real variable $x$ with respect to the Borel probability
measure $||P_{A}(\,)(\psi)||^{2}$. Thus the mean exists, it appears, if and
only if $x$ is integrable with respect to this measure, thus if and only if
$\psi$ is in the domain of $A$. Self-adjoint operators have domains that are
dense in $\mathcal{H}$ but many of the most prominent ones (e.g., those
associated with position and momentum and often energy) do not have domain
equal to $\mathcal{H}$. Hence there will be states for which the means of some
observable may not be well-defined. Whether these states are realizable in
practice is uncertain, but there is no good theoretical reason why they should
be ignored. (Our discussion focuses on mathematical definition and
characterization. The spectrum of a self-adjoint operator is identified with
the possible values of a measured quantity. If the spectrum is discrete, a
measurement may be able to distinguish one value from another; if the spectrum
is continuous, measurement will only be able to determine an interval that
contains the value, not the exact value. Repeated measurements when the system
is in the same state thus only arrive at a rough approximation of the
distribution and a rough estimate of the mean for a state.)

A curiosity in quantum mechanics, not ordinarily seen in other applications of
probability, is the following. Suppose $\mu=\left\langle
A(\psi),\psi\right\rangle$ is the mean of some observable $A$ when the state
is $\psi$. Then the variance of the observable in this state is naturally
given by:

$\int_{\mathcal{R}}(x-\mu)^{2}\,\left\langle
P_{A}(dx)(\psi),\psi\right\rangle\,=\,\left\langle(A-\mu
I)^{2}(\psi),\psi\right\rangle=||(A-\mu I)(\psi)||^{2}.$

So the variance exists if and only if $\psi$ is in the domain of $A$. The
condition for the mean to exist is the same as the condition for the variance
to exist. In quantum mechanics we are led to think that the only distributions
for which the mean is finite are ones in which the variance is also finite.
However, a closer look at this situation reveals some discrepancies.

The chief discrepancy is the following. Suppose that the original observable
$A$ can be written in the form

$A=C-D$

where $C$ and $D$ are non-negative self-adjoint operators. Non-negative self-
adjoint operators can be written as squares of self-adjoint operators, so that
$C=E^{2}$ and $D=F^{2}$ with $E$ and $F$ self-adjoint. Then

$\displaystyle\left\langle
A(\psi),\psi\right\rangle=\left\langle(C-D)(\psi),\psi\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle(E^{2}-F^{2})(\psi),\psi\right\rangle$
$\displaystyle=\left\langle E^{2}(\psi),\psi\right\rangle-\left\langle
F^{2}(\psi),\psi\right\rangle$ $\displaystyle=$
$\displaystyle||E(\psi)||^{2}-||F(\psi)||^{2}.$

Thus, the mean of $A$ exists if and only if $\psi$ is in the intersection of
the domain of $E$ and the domain of $F$. It is easy, incidentally, to
construct examples of elements of $\mathcal{H}$ that are in the domain of a
self-adjoint operator $E$ but are not in the domain of its square $E^{2}$. In
addition, as it happens, it is possible to offer explicit candidates for the
operators $E$ and $F$ given $A$. Set

$E=\int_{(0,\infty)}\sqrt{x}\,P_{A}(dx)\mbox{ and
}F=\int_{(-\infty,0)}\sqrt{-x}\,P_{A}(dx).$

We conclude that the mean of $A$ exists when $\psi$ is in $\mbox{dom
}E\cap\mbox{dom }F$ and the variance of $A$ exists when $\psi$ is in
$\mbox{dom }A$. If $\psi$ is not in $\mbox{dom }E$ but is in $\mbox{dom }F$,
then it is reasonable to say that the mean of $A$ is $\infty$. Likewise if
$\psi$ is in $\mbox{dom }E$ but not in $\mbox{dom }F$, the mean of $A$ is
$-\infty$. If $\psi$ is in neither $\mbox{dom }E$ nor $\mbox{dom }F$, then the
ordinary mean does not exist. In the spirit of our discussion of $L(c)$
earlier, it is possible to truncate the integrals for $E$ and $F$ in the last
display, replacing $\infty$ by $M$ and $-\infty$ by $-M$ and investigate the
existence of an appropriate combined limit as $M$ tend to infinity.

Similar considerations can be applied when the “pure” state $\psi$ is replaced
by a density matrix representing a statistical ensemble of pure states, or in
rigged Hilbert spaces where the existence of states varies according to the
properties of the observable, or to cases arising by the use of positive-
operator-valued measures generalizing the projection-valued measures treated
above.

##  8 Conclusion

The mean, as we have seen, is ubiquitous in scientific explanation. Not only
does it provide a summary of sample data and, when it exists, of data from the
entire population, but it establishes a connection between samples and the
whole population. Furthermore, it facilitates generalization of deterministic
observables that are functions of the deterministic state to probabilistic
observables that are functions of the probabilistic state. The Maximum Entropy
Principle then makes use of constrained means to identify macroscopic
distribution of physical importance parametrized by these means. While quantum
mechanics abandons determinism, it retains the notion of mean to summarize the
possible results of experiments and the measurement of quantum observables.
Although not all observables have finite means, weak and doubly weak means and
the alternatives identified in Theorem 3.1 provide an enumeration of possible
behaviors of variables and associated probability distributions, and give
further insight into potentialities associated with large data sets.

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