Let $M = (\mathbb{R}^n, \mathcal{L})$ be a D1742: Lebesgue measurable space.
Let $\mu^*$ be the D780: Lebesgue outer measure on $\mathbb{R}^n$.
The Lebesgue measure on $M$ is the D992: Function \begin{equation} \mathcal{L} \to [0, \infty], \quad E \mapsto \mu^*(E) \end{equation}