Lebesgue Integration On Euclidean Space

In the same way, when both the Riemann integral and the Lebesgue integral are defined, they give the same value. In related context, I saw several conceptual explanations regarding Lebesgue Integration, but can I see few practical examples that require Lebesgue Integration? What I need is just a toy case of …. Research indicates, Sometimes "Lebesgue theory" is used to refer not only to the tangible Lebesgue integral on $\mathbb R^n$, but also to the abstracted version that uses any $\sigma$-algebra, defines …. Evidence suggests, It's usually easier to define the Lebesgue integral and the supremum of integrating smaller simple functions, but by the Monotone Convergence Theorem, the procedure above …. These findings regarding Lebesgue Integration On Euclidean Space provide comprehensive context for understanding this subject.