# Measure Theory/Measures on Polish spaces

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

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

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

Proof: First, we write

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

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

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

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

Continuing this ping-pong game, we obtain nested sequences ${\displaystyle K_{n_{1},\ldots ,n_{k}}}$ and ${\displaystyle J_{n_{1},\ldots ,n_{k}}}$ dependent on different numbers of indices. Moreover, if we set

${\displaystyle N:=\bigcup _{k\in \mathbb {N} }N_{k}}$ and ${\displaystyle M:=\bigcup _{k\in \mathbb {N} }M_{k}}$,

then ${\displaystyle \mu (N)=\lambda (M)=0}$, so that for any given number of indices, the corresponding sequence of nested sets ${\displaystyle K}$ resp. ${\displaystyle J}$ covers almost all of ${\displaystyle \Omega }$ resp. ${\displaystyle \Psi }$, dependent on whether the number of indices is even or odd. Define ${\displaystyle \Omega ':=\Omega \setminus N}$ and ${\displaystyle \Psi ':=\Psi \setminus M}$. Then define a function ${\displaystyle f:\Omega '\to \Psi '}$ as follows: If ${\displaystyle x\in \Omega '}$, then for any given number of indices ${\displaystyle k}$ there exists one and only one ${\displaystyle K_{n_{1},\ldots ,n_{k}}}$ that contains ${\displaystyle x}$. Moreover, ${\displaystyle \Box }$