dominated convergence theorem

Let $X$ be a measure space, and let $\Phi,f_{1},f_{2},\dots$ be measurable functions such that $\int_{X}\Phi<\infty$ and $|f_{n}|\leq\Phi$ for each $n$. If $f_{n}\rightarrow f$ almost everywhere, then $f$ is integrable and

 $\lim_{n\rightarrow\infty}\int_{X}f_{n}=\int_{X}f.$

This theorem is a corollary of the Fatou-Lebesgue theorem.

Title dominated convergence theorem DominatedConvergenceTheorem 2013-03-22 13:12:47 2013-03-22 13:12:47 Koro (127) Koro (127) 13 Koro (127) Theorem msc 28A20 Lebesgue’s dominated convergence theorem MonotoneConvergenceTheorem FatousLemma VitaliConvergenceTheorem