Let $(X,\mathcal{F},\mu)$ be a Measure Space, let $E\in\mathcal{F}$. Let $f:X\to \mathbb{R}^{+}$ be a Simple Function: $$f = \sum_{i=1}^{n}a_{i}\mathbb{1}_{A_{i}}$$ where $A_{i}=f^{-1}(\{ \alpha_{i} \}),\ \forall_{1}\le i\le n$. The Lebesgue Integral of $f$ with respect to measure $\mu$ is defined as: $$\int\limits _{E}f \, d\mu =\sum_{i=1}^{n}a_{i} \ \mu(A_{i}\cap E)$$where $A_{i}\in\mathcal{F},a_{i}\ge0$.

We can further the definition of where for $f:X\to[0,\infty)$ measurable we define the Lebesgue Integral as: $$\int\limits _{X}f \, d\mu =\sup\left\{ \int\limits _{X}g \, d\mu :0\le g<f, \ g \ \text{simple} \right\}$$

Finally, adding on to the Integral of Positive Measurable Functions, let $f:X\to \mathbb{R}$ be measurable. Let $f^{+}=max(f,0)$, $f^{-}=max(-f,0)$ we define the Lebesgue Integral as: $$\int\limits _{X}f \, d\mu=\int\limits _{X}f^+ \, d\mu-\int\limits _{X}f^- \, d\mu$$

If $f:X\to \mathbb{C}$ is s.t. $f=u+iv$ and $u,v:X\to \mathbb{R}$ are measurable functions, and if $f\in\mathscr{L}^{1}(X,\mathscr{M},\mu)$ then $$\int\limits _{E}f \, d\mu=\int\limits _{E}u^{+} \, d\mu -\int\limits _{E}u^{-} \, d\mu+i\int\limits_{E}v^{+}\,d\mu-i \int\limits _{E}v^{-} \, d\mu$$for all $E\in\mathscr{M}$. ## Note: $$f\text{ measurable}\implies f^{+},f^{-}\text{ measurable}$$by Composition of Measurable Functions. This is only possible with Borel σ-algebra hence why we restrict ourselves to this instead of Lebesgue Measurable. ## Properties 1. Linearity: $$I(\alpha f+\beta g)=\alpha I(f)+\beta I(g)$$ 2. Monotonicity: $$f\le g\implies I(f)\le I(g)$$