Theorem (Lebesgue Measurable σ-algebra)

Let $\lambda^{*}:\mathcal{P}(X)\to[0,\infty]$ is an Outer Measure, then:

• $\mathcal{M}(\lambda^{*}):=\{ A\subset X:A \ \ \lambda^{*}$-measurable$\}$ is a σ-algebra.
• $\lambda:\mathcal{M}(\lambda^{*})\to[0,\infty]$ is a measure.