8.2. Uniform Convergence

Theorem 8.2.10: Lebesgue's Bounded Convergence Theorem

Let { fn } be a sequence of (Lebesgue) integrable functions that converges almost everywhere to a measurable function f. If |fn(x)| g(x) almost everywhere and g is (Lebesgue) integrable, then f is also (Lebesgue) integrable and:
| fn - f | dm = 0

Please refer to any standard Graduate-level textbook on Analysis for the - involved - proof.
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