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 $\lim_{k\to\infty}\int_E |f_k-f|=0$ .