Suppose is a sequence of non-negative measurable functions, .
Define for all .
By definition, and , so by Monotone Convergence Theorem,
Furthermore, by definition we have , implying that .
Since exists, taking of both sides yields:
- ↑ Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, §2.2
- ↑ Craig, Katy. MATH 201A Lecture 14. UC Santa Barbara, Fall 2020.