ACIT4330-Page/content/Revision/Real Analysis/Question 15.md

Question

State Lebesgue's monotone convergence theorem.

Answer

Say X has a Measure \mu, let f_{n} : X \to [0, \infty] be Measurable, and f_{1} \leq f_{2} \leq f_{3} \leq \dots Then

\int f_{m} \, d\mu \to \int \lim_{ n \to \infty } f_{n} \, d\mu

as m \to \infty