Definition:Translation Invariant Measure

Definition

Let $\mu$ be a measure on $\R^n$ equipped with the Borel $\sigma$-algebra $\map \BB {\R^n}$.

Then $\mu$ is said to be translation invariant or invariant under translations if and only if:

$\forall x \in \R^n, \forall B \in \map \BB {\R^n}: \map \mu {x + B} = \map \mu B$

where $x + B$ is the set $\set {x + b: b \in B}$.

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