Measure Theory/Measures on Polish spaces

Theorem (von Neumann's big theorem on algebra maps):

Let $\mathrm{\Omega },\mathrm{\Psi }$ be Polish spaces that are part of the outer regular measure spaces $(\mathrm{\Omega },ℱ,\mu )$ and $(\mathrm{\Psi },𝒢,\lambda )$. Suppose that $\varphi :ℱ\to 𝒢$ is an algebra map such that for all $D\in 𝒢$, there exists an $A\in ℱ$ such that $\lambda (\varphi (A)\mathrm{\Delta }D)=0$. Then there exists ${\mathrm{\Omega }}^{\prime }\subseteq \mathrm{\Omega }$ and ${\mathrm{\Psi }}^{\prime }\subseteq \mathrm{\Psi }$ such that $\mu (\mathrm{\Omega }\setminus {\mathrm{\Omega }}^{\prime })=\lambda (\mathrm{\Psi }\setminus {\mathrm{\Psi }}^{\prime })=0$ and a bijective function $f:{\mathrm{\Omega }}^{\prime }\to {\mathrm{\Psi }}^{\prime }$ such that

1. for $A\in ℱ$, $\lambda (f(A\cap {\mathrm{\Omega }}^{\prime })\mathrm{\Delta }\varphi (A))=0$ and
2. for $D\in 𝒢$, $\mu (f^{-1}(D\cap {\mathrm{\Psi }}^{\prime })\delta A)=0$, where $A\in ℱ$ is any set such that $\lambda (\varphi (A)\mathrm{\Delta }D)=0$.

Proof: First, we write

$\mathrm{\Omega }=\underset{n\in \mathbb{N}}{⨆}{K}_n\cup N_1$,

where $N_1$ is a nullset and each $K_j$ is closed and of diameter $\le 1$. Then write ${J_j}^{\prime }:=\varphi (K_j)$ and $J_j:=\overline{{J_j}^{\prime }}$. Then we write

$\mathrm{\Psi }=\underset{n_1,n_2\in \mathbb{N}}{⨆}{J}_{n_1,n_2}\cup M_2$,

where $J_{n_1,n_2}\subseteq J_{n_1}$ is closed and each $J_{n_1,n_2}$ has diameter $\le 1/2$. Then set $K_{{n_1}^{\prime },n_2}$ to be a set such that $\lambda (\varphi (K_{{n_1}^{\prime },n_2})\mathrm{\Delta }{J}_{n_1,n_2})=0$ and $K_{n_1,n_2}$ its closure.

Continuing this ping-pong game, we obtain nested sequences $K_{n_1,\dots ,n_k}$ and $J_{n_1,\dots ,n_k}$ dependent on different numbers of indices. Moreover, if we set

$N:=\bigcup _{k\in \mathbb{N}}{N}_k$ and $M:=\bigcup _{k\in \mathbb{N}}{M}_k$,

then $\mu (N)=\lambda (M)=0$, so that for any given number of indices, the corresponding sequence of nested sets $K$ resp. $J$ covers almost all of $\mathrm{\Omega }$ resp. $\mathrm{\Psi }$, dependent on whether the number of indices is even or odd. Define ${\mathrm{\Omega }}^{\prime }:=\mathrm{\Omega }\setminus N$ and ${\mathrm{\Psi }}^{\prime }:=\mathrm{\Psi }\setminus M$. Then define a function $f:{\mathrm{\Omega }}^{\prime }\to {\mathrm{\Psi }}^{\prime }$ as follows: If $x\in {\mathrm{\Omega }}^{\prime }$, then for any given number of indices $k$ there exists one and only one $K_{n_1,\dots ,n_k}$ that contains $x$. Moreover, $◻$