Lebesgue Measure is Invariant under Translations

Theorem
Let $\lambda^n$ be the $n$-dimensional Lebesgue measure on $\R^n$ equipped with the Borel $\sigma$-algebra $\mathcal B \left({\R^n}\right)$.

Let $\mathbf x \in \R^n$.

Then $\lambda^n$ is translation-invariant; i.e., for all $B \in \mathcal B \left({\R^n}\right)$, have:


 * $\lambda^n \left({\mathbf x + B}\right) = \lambda^n \left({B}\right)$

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

Note
This proof formalizes the physical intuition that the size of an object does not depend on its position.