Theorem (4) Section 1.1. Walters' Ergodic Theory

Question

The following is from Walters' Ergodic Theory book :

I have studied Measure Theory but I have no knowledge of Haar measure more than what I know from here. I can't understand some parts of the proof of Theorem (4) and its following example for that :

1- Why $m$ is regular? (the answer is in this theorem below which I couldn't prove it) :

(any easy for self-study reference for proving Theorem 0.13 including a friendly introduction to Haar measure would be much appreciated)

2- How regularity of $m$ implies regularity of $\mu$?

3- How $m(A^{-1}(Ax • E)) = m(x • A^{-1} E)$ holds? (I know $m(x • E) = m(E)$.)

4- How $T(z)=z^n$ is measure preserving? Roughly speaking $m(T^{-1}(B))=m(B)$ is not consistent with what happens with $T$ which 'expands' sets by a factor of $n$ for example intervals on a unit circle so not a measure preserving.

Answer 1

1 As you said the Theorem proving the existsence of the Haar measure gives regularity. To understand why this holds, you must read the proof of Theorem 0.13.

2 Let $E \subset G$ be any Borel set. Then $$ \mu(E)=m(A^{-1}E)= \sup \{ m(K) : K\subseteq A^{-1}E , K \mbox{ compact } \} $$ Now use the fact that $A$ is onto and $G$ is compact to show that each compact $ K\subseteq A^{-1}E$ can be written in the form $K=A^{-1}K'$ for some $K' \subseteq E$. This gives $$ \mu(E)=m(A^{-1}E)= \sup \{ m(A^{-1}K') : K'\subseteq E , K' \mbox{ compact } \} = \sup \{ \mu(K') : K'\subseteq E , K' \mbox{ compact } \} $$ which is the inner regularity of $\mu$.

The outher regularity is proven similarly.

3 This is about sets not measures. $$ A^{-1}(Ax\cdot E)= \{ y : Ay \in Ax\cdot E \}= \{ y : \exists z \in E Ay= Ax\cdot z \}= \{ y : \exists z \in E Ax^{-1}\cdot Ay= z \}= \{ y : \exists zz \in E A(x^{-1}\cdot y) =z \}= \{ y : \exists z \in, x^{-1}\cdot y = A^{-1}z \}= \{ y : \exists z \in, y = x\cdot A^{-1}z \} =x \cdot A^{-1}E $$

4 You are pulling back intervals, not pushing them forward.

Roughly speaking, if $E$ is an interval of lenght $\epsilon$, then $T^{-1}(E)$ consists of $n$ intervals of lenght $\frac{\epsilon}{n}$, which gives the measure preserving. The fact that you get $n$ intervals compensates for the fact that $T^{-1}$ contracts by a factor of $n$.

This is exactly why the definition uses $T^{-1}$ to pull back the measure, and not $T$.