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 :


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

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