distance. More precisely, this distance can be explicitly computed in terms of pseudo-inverses.

\[def:cdf-pseudo-inverse\] Given a non-negative finite measure $\mu\in{\mathfrak{M}_+}({\mathbb{R}})$ on the real line, we define its *cumulative distribution function (CDF)* $F_\mu:{\mathbb{R}}\to[0,\mu({\mathbb{R}})]$ by $$F_\mu(x) := \mu((-\infty,\,x]),$$ and its *pseudo-inverse* $X_\mu:[0,\mu({\mathbb{R}}))\to{\mathbb{R}}$ by $$X_\mu(z) := \inf \left\{x \in \mathbb{R} : F_\mu(x) > z\right\}.$$

By definition, $F_\mu$ is a càdlàg function, and so is $X_\mu$, i.e., $X_\mu\in{\mathrm{D}}([0,\mu({\mathbb{R}})))$. Note that in some cases, $X_\mu$ indeed is an inverse of $F_\mu$. Namely, if $F_\mu$ is *strictly* monotonically increasing, corresponding to $\mu$ having its support on the whole of $\mathbb{R}$, then $X_\mu \circ F_\mu = \mathrm{id}$. If $F$ is *continuous*, which means that $\mu$ does not give mass to points, then $F_\mu\circ X_\mu = \mathrm{id}$. However, in general we only have $$\label{eq:219}
  (X_\mu\circ F_\mu) (x) \geq x, \ x \in \mathbb{R}, \quad (F_\mu\circ X_\mu)(z) \geq z, \ z\in[0,\mu({\mathbb{R}})).$$

\[lem:16\] Given $\nu\in{\mathfrak{M}_+}({\mathbb{R}})$, then, for all $f \in L^1({\mathbb{R}};\nu)$, $$\int_\mathbb{R} f(x) \, \mathrm{d} \nu(x) = \int_0^{\nu({\mathbb{R}})} f(X(z)) \, \mathrm{d} z.$$

A direct consequence of the substitution formula is the following convenient representation for the convolution of functions with measures.

If $\psi:{\mathbb{R}}\to{\mathbb{R}}$ is a continuous function with at most quadratic growth of $|\psi(x)|$ for $x\to\pm\infty$, and if $\nu\in{\mathcal{P}}_2({\mathbb{R}})$ has 