quasi-invariant

Definition 1.

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

As a example, let $E=ℝ$ with $ℬ$ the Borel $\sigma$-algebra (http://planetmath.org/BorelSigmaAlgebra), 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 $\mu (T^{-1}(\{0\}))=\mu (ℝ)=\mathrm{\infty }$).

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