ACIT4330-Page/content/Revision/Real Analysis/Question 14.md

Question

Define the Lebesgue integral of a extended non-negative measurable function.

Answer

The Lebesgue Integral, A \in M of a Measurable function f : X \to [0, \infty] is

\int_{A} f \, d\mu \equiv \sup_{0\leq s\leq f} \int_{A} s \, d\mu \in [0, \infty]

[!info] What is s? s is Measurable and a Simple Function

That is the simple function brings: s : X \to \mathbb{R} of the form s = \sum_{i=1}^{n} a_{i} \times X_{a_{i}} for pairwise disjoint A_{i} \subset X