Let $\mathscr{A}\subseteq 2^{X}$ be an algebra, and let $\lambda:\mathscr{A}\to[0,+\infty]$ be a pre-measure. Let $\mu^{*}$ be defined $\forall E\subseteq X$ as $$\mu^{*}(E)=\inf\left\{ \sum_{j=1}^{\infty}\lambda(A_{j}):A(j)_{j\ge 1}\subseteq \mathscr{A}\text{ s.t. }E\subseteq \bigcup_{j=1}^{\infty}A_{j} \right\}$$(which by Lebesgue Outer Measure is an outer measure). Let $\mathscr{M}$ be the σ-algebra of $\mu^{*}$-measurable sets. Then 1. $\left.\mu^{*}\right|_{\mathscr{A}}=\lambda$ 2. $\mathscr{A}\subseteq \mathscr{M}$