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Let $(X,\mathcal{S},\mu)$ be a $\sigma$-finite measure space and $\phi:X\rightarrow X$ be a measurable transformation (in the sense that $\phi^{-1}(A)\in\mathcal{S}$ for each $A\in\mathcal{S}$). For each $n\geq1$, put $\mathcal{S}_n=\phi_n^{-1}(\mathcal{S})$ and $\mathcal{S}'=\cap_{n\geq1}\mathcal{S}_n$, where $\phi_n=\phi\circ\phi\circ\dots\circ\phi$ $n$-times. Is $(X,\mathcal{S}',\mu)$ a $\sigma$-finite measure space?

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Consider $X:=\mathbf R$, $\mathcal S$ the Borel $\sigma$-algebra and $\mu$ the Lebesgue measure. Let $\phi$ be the characteristic function of a Borel subset $A$ such that:

  • $\mu(A)=+\infty$,
  • $1\in A$;
  • $0\notin A$.

Then $\phi_n(x)=\chi_A(x)$ for each $n\geqslant 1$. Consequently, $\mathcal S'=\{\emptyset,A,\mathbb R\setminus A,\mathbb R\}$ and the Lebesgue measure is not $\sigma$-finite over this $\sigma$-algebra..

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