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# quasi-invariant

###### Definition 1.

Let $(E,\mathcal{B})$ be a measurable space, and $T:E\to E$ be a measurable map. A measure $\mu$ on $(E,\mathcal{B})$ is said to be *quasi-invariant* under $T$ if $\mu\circ T^{{-1}}$ is absolutely continuous with respect to $\mu$. That is, for all $A\in\mathcal{B}$ with $\mu(A)=0$, we also have $\mu(T^{{-1}}(A))=0$. We also say that $T$ leaves $\mu$ quasi-invariant.

As a simple example, let $E=\mathbb{R}$ with $\mathcal{B}$ the Borel $\sigma$-algebra, and $\mu$ be Lebesgue measure. If $T(x)=x+5$, then $\mu$ is quasi-invariant under $T$. If $S(x)=0$, then $\mu$ is not quasi-invariant under $S$. (We have $\mu(\{0\})=0$, but $\mu(T^{{-1}}(\{0\}))=\mu(\mathbb{R})=\infty$).

To give another example, take $E$ to be the nonnegative integers and declare every subset of $E$ to be a measurable set. Fix $\lambda>0$. Let $\mu(\{n\})=\frac{\lambda^{n}}{n!}$ and extend $\mu$ to all subsets by additivity. Let $T$ be the shift function: $n\to n+1$. Then $\mu$ is quasi-invariant under $T$ and not invariant.

## Mathematics Subject Classification

28A12*no label found*

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