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I have the following statement from the first Chapter "Couplings and changes of variables" of the book Optimal Transport: Old and New by Cédric Villani

The increasing rearrangement on $\mathbb{R}$. Let $\mu, \nu$ be two probability measures on $\mathbb{R}$; define their cumulative distribution functions by $$ F(x)=\int_{-\infty}^x d \mu, \quad G(y)=\int_{-\infty}^y d \nu . $$ Further define their right-continuous inverses by $$ \begin{aligned} &F^{-1}(t)=\inf \{x \in \mathbb{R} ; \quad F(x)>t\} \\ &G^{-1}(t)=\inf \{y \in \mathbb{R} ; \quad G(y)>t\} \end{aligned} $$ and set $$ T=G^{-1} \circ F . $$ If $\mu$ does not have atoms, then $T_{\#} \mu=\nu$. This rearrangement is quite simple, explicit, as smooth as can be, and enjoys good geometric properties.

Why is that "If $\mu$ does not have atoms, then $T_{\#} \mu=\nu$"? Why do we need that $\mu$ does not have atoms?

Any help is appreciated.

Tab1e
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I will use the left-continuous inverse $H^\leftarrow(p)=\inf\{x:H(x)\geq p\}$.

We know that if $Z$ has cdf $H$ (which is right-continuous and nondecreasing) and $U\sim \textrm{Uniform}(0,1)$ then:

  1. if $H$ is continuous, then $H(Z)\sim U$;
  2. $H^\leftarrow (x)\leq y \iff H(y)\geq x$

You can find these facts in McNeil et al. (Proposition 5.2.(2) and Proposition A.3.(iv)).

If $X\sim \mu$ and it does not have atoms, then this implies it has a continuous cdf $F$. We can state: $$P(T(X)\leq y)=P(G^{\leftarrow}(F(X))\leq y)=P(F(X)\leq G(y))=G(y)$$ Now since $P(T(X)\leq y)=(T_\#\mu)((-\infty,y])$ and the measures $T_\#\mu,\nu$ agree on an $\cap$-stable generator of the Borel sets on the real line, we can state that $T_\#\mu=\nu$ on $(\mathbb{R},\mathscr{B}(\mathbb{R}))$.

Snoop
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  • Thank you for your answer! I am not quite familiar with this area, so please allow me to ask some questions to make sure that I really understand: So the non-atomic property implies the continuity of the CDF of F which gives us $F(X)\sim U$. But how do you get the last equality? ($P(U\leq G(y))=G(y)$) – Tab1e Nov 29 '22 at 19:47
  • The last equality is the value of the cdf of a uniform rv on $(0,1)$ at point $G(y)$ @Tab1e – Snoop Nov 29 '22 at 19:51