Definition:Translation Invariant Measure

Definition
Let $\mu$ be a measure on $\R^n$ equipped with the Borel $\sigma$-algebra $\mathcal B \left({\R^n}\right)$.

Then $\mu$ is said to be translation-invariant or invariant under translations iff:


 * $\forall x \in \R^n, \forall B \in \mathcal B: \mu \left({x + B}\right) = \mu \left({B}\right)$

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