ACIT4330-Page/content/Definitions/Measure Theory/Lebesgue Integral.md

The **Lebesgue Integral** (over $A \in M$) of [[Measurable|measurable]] $f : X \to [0, \infty]$ is $\int_{A} f \, d\mu \equiv \underbrace{\sup_{0 \leq s \leq f}}_{s\ \text{measurable and simple}} \int_{A} s \, d\mu \in [0, \infty]$.