# Measure Theory/Measures on Polish spaces

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

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

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

Proof: First, we write

$\Omega =\bigsqcup _{n\in \mathbb {N} }K_{n}\cup N_{1}$ ,

where $N_{1}$ is a nullset and each $K_{j}$ is closed and of diameter $\leq 1$ . Then write $J_{j}':=\phi (K_{j})$ and $J_{j}:={\overline {J_{j}'}}$ . Then we write

$\Psi =\bigsqcup _{n_{1},n_{2}\in \mathbb {N} }J_{n_{1},n_{2}}\cup M_{2}$ ,

where $J_{n_{1},n_{2}}\subseteq J_{n_{1}}$ is closed and each $J_{n_{1},n_{2}}$ has diameter $\leq 1/2$ . Then set $K_{n_{1},n_{2}}'$ to be a set such that $\lambda (\phi (K_{n_{1},n_{2}}')\Delta J_{n_{1},n_{2}})=0$ and $K_{n_{1},n_{2}}$ its closure.

Continuing this ping-pong game, we obtain nested sequences $K_{n_{1},\ldots ,n_{k}}$ and $J_{n_{1},\ldots ,n_{k}}$ 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 $\Omega$ resp. $\Psi$ , dependent on whether the number of indices is even or odd. Define $\Omega ':=\Omega \setminus N$ and $\Psi ':=\Psi \setminus M$ . Then define a function $f:\Omega '\to \Psi '$ as follows: If $x\in \Omega '$ , then for any given number of indices $k$ there exists one and only one $K_{n_{1},\ldots ,n_{k}}$ that contains $x$ . Moreover, $\Box$ 