Lebesgue Integral

Definition

Riemann integral and Lebesgue integral

Consider a measure spaces $(\Omega ,\mathcal{F},\mu )$ and a Measure Space $(\overline{\mathbb{R}},\mathcal{R})$, where $\mathcal{R}$ is the Borel Sigma-Field on $\overline{\mathbb{R}}$, and a Measurable Function $f:(\Omega ,\mathcal{F})\to (\overline{\mathbb{R}},\mathcal{R})$. Then the Lebesgue integral of the function $f$ with respect to $\mu$ is denoted as:

${\int }_{\Omega }fd\mu ={\int }_{\Omega }f(x)d\mu (x)={\int }_{\Omega }f(x)\mu (dx)$

Approximation of Lebesgue Integral with Finite Sum

$\int fd\lambda \approx \sum _{i=1}^N{a}_n\lambda (A_n)$ where $A_n:=\{x|f(x)=a_n\}$ or $A_n=f^{-1}(\{a_n\})$.

To compute the Riemann Integral of $f$, one partitions the domain $[a,b]$ into sub-intervals, while in the Lebesgue integral, one partitions the image of $f$.