Generalized Lebesgue Dominated Convergence Theorem Proof

This key theorem showcases the full power of Lebesgue Integration Theory.

Generalized Lebesgue Dominated Convergence Theorem

Let \{f_k\} and \{\phi_k\} be sequences of measurable functions on E satisfying f_k\to f a.e. in E, \phi_k\to \phi a.e. in E, and |f_k|\leq\phi_k a.e. in E. If \phi\in L(E) and \int_E \phi_k\to\int_E \phi, then \int_E |f_k-f|\to 0.

Proof

We have |f_k-f|\leq|f_k|+|f|\leq\phi_k+\phi. Applying Fatou’s lemma to the non-negative sequence \displaystyle h_k=\phi_k+\phi-|f_k-f|, we get \displaystyle 2\int_E\phi\leq\liminf_{k\to\infty}\int_E (\phi_k+\phi-|f_k-f|).
That is, \displaystyle 2\int_E \phi\leq2\int_E\phi-\limsup_{k\to\infty}\int_E |f_k-f|.

Since \int_E\phi<\infty, we get \limsup_{k\to\infty}\int_E |f_k-f|\leq 0. Since \liminf_{k\to\infty}\int_E |f_k-f|\geq 0, this implies \lim_{k\to\infty}\int_E |f_k-f|=0.

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