# Measurable sets on Polish spaces

A measure on a Polish space $X$ will be refered as a function $\mu:BOREL(X)\rightarrow [0,1]$ s.t.

1. $\mu(\emptyset)=0,\mu(X)=1$

2. If $\{A_n:n\in\omega\}\subseteq BOREL(X)$ is a sequence of pairwise disjoint sets, then $\mu(\cup_{n\in\omega}A_n)=\sum_{n\in\omega}A_n$

3. $\mu$ is nonatomic: for every set $A\in BOREL(X)$ s.t. $\mu(A)>0$ there exists $B\in BOREL(X)$ s.t. $\mu(A)>\mu(B)>0$.

4. $\mu$ is translation invariant: for $A\in BOREL(X)$ and $t\in X$, $\mu(t+A)=\mu(A)$ (if X is a topological group)

5. for every $A\in BOREL(X)$ and $\epsilon>0$, there exists a compact set and an open set $U$ such that $K\subseteq A \subseteq U$ und $\mu(U\cap K^{c})<\epsilon$

When $\mu$ is a measure, it can be extended to a $\sigma$-algebra as $MEASURABLE(X)=\{A:\exists B\in BOREL(X) \mu(A\triangle B)=0\}$

I am not familiar with this kind of definition of measurable sets. I attended a basic course in measure theory which introduced the a measure on the real line as using algebras,... and outer measures $\lambda^*$ to define measurable $A$ sets via the equation $\lambda^*(E\cap A)+\lambda^*(E\cap A^c)=\lambda(E)$ for all $E\subseteq \mathbb{R}$.

It first sight, i am not able to see the connection here. Is there a good reference for the first way to introduce measurable sets.

• $\mu$ is not defined on $X$ but on some subset of $\mathscr{P}(X)$, presumably the Borel sets. – Henno Brandsma May 28 '16 at 11:29
• A Polish space need not have an addition. – Henno Brandsma May 28 '16 at 12:17
• In the last definition why is the measure of $A \triangle B$ defined at all? – Henno Brandsma May 28 '16 at 12:19
• $BOREL(X)$ is the smallest $\sigma$-algebra containing all open sets. The book says, when "$\mu$ is a measure, it can be extended to a $\sigma$-algebra $MEASURABLE(X)=\{A:\exists B\in BOREL(X)\mu (A\triangle B)=0\}$". Not sure, what "it" does refer to, since $BOREL(X)$ is already a $\sigma$-algebra. – peer May 28 '16 at 13:56

This definition is inexact: it should probably be defined on $\text{BOREL}(X)$ (by property 2, and the fact that all compact and open sets are measurable).
The property non-atomic is not defined (it probably means there are no atoms, where an atom is a Borel (?) set $A$ such that $\mu(A) > 0$ and for all subsets $B$ of $A$ either $\mu(B) = \mu(A)$ or $\mu(B) = 0$).
Translation invariance is also undefined. It might mean that if $h$ is an isometry of $X$ and $A$ is part of the domain of $\mu$, then $\mu(A) = \mu(h[A])$.
• It's a book about the real line and polish spaces equipped with $\sigma$-additive measures satisfying the above properties (for example the lebesgue-measure on the real-line). – peer May 28 '16 at 14:09