## 8.2. Uniform Convergence

### Theorem 8.2.10: Lebesgue's Bounded Convergence Theorem

Let

Please refer to any standard Graduate-level textbook on Analysis for
the - involved - proof.
*{ f*be a sequence of (Lebesgue) integrable functions that converges almost everywhere to a measurable function_{n}}*f*. If*|f*almost everywhere and_{n}(x)| g(x)*g*is (Lebesgue) integrable, then*f*is also (Lebesgue) integrable and:| f_{n}- f | dm = 0