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.

  • $\begingroup$ $\mu$ is not defined on $X$ but on some subset of $\mathscr{P}(X)$, presumably the Borel sets. $\endgroup$ – Henno Brandsma May 28 '16 at 11:29
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    $\begingroup$ A Polish space need not have an addition. $\endgroup$ – Henno Brandsma May 28 '16 at 12:17
  • $\begingroup$ In the last definition why is the measure of $A \triangle B$ defined at all? $\endgroup$ – Henno Brandsma May 28 '16 at 12:19
  • $\begingroup$ $BOREL(X)$ 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. $\endgroup$ – 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 $\mu(A) = \mu(h[A])$.

So this text only considers Radon probability measures with special properties, so this is much more specialised than general measure theory. But any good measure theory book (Halmos, or Fremlin's books, or others) will prove that for such specialised measures the measurable subsets in the general sense will coincide with this definition.

  • $\begingroup$ I did some corrections. $\endgroup$ – peer May 28 '16 at 12:15
  • $\begingroup$ 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). $\endgroup$ – peer May 28 '16 at 14:09

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