Definition (Lebesgue measurable (891))

Suppose $\mu^{*}$ is an Outer Measure on $X$. We say $A\subseteq X$ is $\mu^{*}$-measurable if and only if $$\forall E\subseteq X:\mu^{*}(E)=\mu^{*}(E\cap A)+\mu^{*}(E\cap A^{c})$$

Definition (Lebesgue measurable (437))

Let $\lambda^{*}:\mathcal{P}(X)\to[0,\infty]$ denote the Lebesgue Outer Measure on $X$, and let $A\subset X$. Then $A$ is $\lambda^{*}$-measurable or Carathéodory-measurable or Lebesgue-measurable if and only if $$\lambda^{*}(E)=\lambda^{*}(E\cap A)+\lambda^{*}(E\cap A^{c})$$$\forall E\subset X$. We denote the σ-algebra of Lebesgue Measurable sets as $\mathcal{M}(\lambda^*)$.