|
|
|
|
quasi-invariant
|
(Definition)
|
|
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.
|
"quasi-invariant" is owned by Mathprof. [ full author list (2) | owner history (2) ]
|
|
(view preamble | get metadata)
Cross-references: function, additivity, fix, measurable set, subset, integers, Lebesgue measure, absolutely continuous, measure, map, measurable, measurable space
There are 2 references to this entry.
This is version 9 of quasi-invariant, born on 2006-05-31, modified 2006-10-07.
Object id is 7942, canonical name is QuasiInvariant.
Accessed 1148 times total.
Classification:
| AMS MSC: | 28A12 (Measure and integration :: Classical measure theory :: Contents, measures, outer measures, capacities) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
