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I'm having problems with this homework question on Riesz Representation theorem and was wondering if anyone could give me some advice? Unfortunately, as an Economist taking a measure theory class I'm not as well prepared as I should be. So I'm finding this question rather challenging. Hopefully I'll improve my maths skills by taking this course.

$\textbf{Question}$

Let $X$ be a compact metric space and let $\mathcal{B}(X)$ denote the Borel $\sigma$-algebra on X. Also let $(M,\mathcal{M},\mu)$ be a measure space with $\mu(M)<\infty$. Now let $\phi: M \rightarrow X$ be a $\mathcal{M}−\mathcal{B}(X)$-measurable map and define $\Phi:C(X)\rightarrow \mathbb{R}$ by \begin{equation} \Phi(f)=\int f \circ \phi \,d\mu\;\;\text{for all}\;\;f\in C(X) \end{equation}

It follows from above and the Riesz Representation Theorem for positive functionals that there is a unique finite Borel measure $\mu_{\Phi}$ on $X$ such that \begin{equation} \Phi(f)=\int f \,d\mu_{\Phi}\;\;\text{for all}\;\;f\in C(X) \end{equation}

Define $\mu_{\phi}:\mathcal{B}(X)\rightarrow [0,\infty]$ by $\mu_{\phi}(B)=\mu(\phi^{-1}(B))$ for $B \in \mathcal{B}(X)$.

(iii) Show that $\mu_{\Phi}=\mu_{\phi}$.

[Hint: Recall that if $f:M\rightarrow [0,\infty]$ is a positive Borel function then there is an increasing sequence $(s_n)_n$ of positive and simple Borel functions $s_n:M\rightarrow [0,\infty]$ such that $s_n \nearrow f$ pointwise.]

I have a strategy that I'm confident will work. If I can show that $\Phi(f)$ is represented by the two integrals above, by the uniqueness part of the Riesz Representation theorem we can conclude that $\mu_{\Phi}=\mu_{\phi}$ (I think). From the hint, I know we need to first consider simple functions or characteristic functions and show they are the same on all Borel sets then use a convergence theorem to show that the simple functions converge to $f$. However, I don't know the mechanics of how to do this. It should be trivial (or some I'm told) to show this for characteristic functions and simple functions but I don't see how.

Many thanks for your help.

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  • $\begingroup$ Hint: use the characteristic function of a set $B$ to compute $\mu_{\Phi}(B)$. $\endgroup$ Jan 10, 2016 at 15:48
  • $\begingroup$ I get $1_{\phi^{-1}(B)}$ when I do that but I can't get further. Thanks for the comment. $\endgroup$
    – mark
    Jan 10, 2016 at 16:04

1 Answer 1

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Let $E$ be a measurable set, $\mu_{\Phi}(E)=\int 1_Edu_{\Phi}=\int 1_E\circ \phi d\mu=\int 1_{\phi^{-1}(B)}d\mu=\mu(\phi^{-1}(B))$.

Remark that $(1_E\circ\phi)(x)=1$ iff $1_E(\phi(x))=1$ this equivalent to saying that $\phi(x)\in E$ or equivalently $x\in \phi^{-1}(E)$ so $1_E\circ \phi =1_{\phi^{-1}(E)}$.

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  • $\begingroup$ My problem is that I don't see why $\int 1_E \circ \phi d\mu=\int 1_{\phi^{-1}(B)} d\mu$. But I'll go away and think about it. Thanks for your help. $\endgroup$
    – mark
    Jan 10, 2016 at 16:16

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