Definition (891)

The Lebesgue measure, $m$, is defined as the special case of the Lebesgue-Stieltjes Measure where $F:\mathbb{R}\to \mathbb{R}$ is defined as $$F(x)=x\quad\forall x \in\mathbb{R}$$ # Definition (437) The Lebesgue Measure, $\lambda$ is the restriction of the Lebesgue Outer Measure, $\lambda^{*}$, to the σ-algebra of Lebesgue measurable sets i.e. $$\lambda(A)=\lambda^{*}(A), \ \forall A\in\mathcal{M}(\lambda^{*})$$we generally define the Lebesgue measure in the 1D sense as $$\lambda([a,b])=b-a$$ ## Note The Lebesgue measure is σ-finite.