Lebesgue Integral of Non-Negative Function

Definition

Consider a measure spaces $(\Omega ,\mathcal{F},\mu )$ and a Measure Space $(\overline{\mathbb{R}},\mathcal{R})$ and a positive 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 defined as ${\int }_{\Omega }fd\mu :=\sup \left\{{\int }_{\Omega }\phi d\mu \right|0\le \phi \le f,\text{a.e. w.r.t.}\mu \}$ where each $\phi :(\Omega ,\mathcal{F})\to (\mathbb{R},\mathcal{R})$ is a Simple Function.

Facts

$\int fd\mu$ can be infinity