Lebesgue Integral of Simple Function

Definition

Consider a measure spaces $(\Omega ,\mathcal{F},\mu )$ and a Measure Space $(\overline{\mathbb{R}},\mathcal{R})$, and a positive $\mathcal{F}-\mathcal{R}$ measurable Simple Function, $f:(\Omega ,\mathcal{F})\to (\overline{\mathbb{R}},\mathcal{R})$, with Codomain $R:=\{r_1,r_2,\dots ,r_n\}$ that have corresponding preimages $f^{-1}(r_i)=A_i\in \mathcal{F}$. Then the Lebesgue integral of the function $f$ with respect to $\mu$ is defined as ${\int }_{\Omega }fd\mu ={\sum }_{i=1}^n{r}_i\mu (A_i)$

Facts

Consider a measure spaces $(\Omega ,\mathcal{F},\mu )$ and a Measure Space $(\overline{\mathbb{R}},\mathcal{R})$, and a positive $\mathcal{F}-\mathcal{R}$ measurable Simple Function, $f:(\Omega ,\mathcal{F})\to (\overline{\mathbb{R}},\mathcal{R})$, with Codomain $R:=\{r_1,r_2,\dots ,r_n\}$ that have corresponding preimages $f^{-1}(r_i)=A_i\in \mathcal{F}$. Then, $f={\sum }_{i=1}^m{\alpha }_i{1}_{A_i}={\sum }_{j=1}^n{\beta }_j{1}_{B_j}⟹\int fd\mu ={\sum }_{i=1}^n{\alpha }_i\mu (A_i)={\sum }_{i=1}^n{\beta }_i\mu (B_i)$