# Convergence Theorems

For all statements on this page, assume that $(X,A,μ)$ is a measure space.

Monotone Convergence Theorem

For each $n∈N$ let $f_{n}:X→R$ be a measurable function. If $0≤f_{n}≤f_{n+1}$ almost everywhere, then

$∫_{X}n→∞lim f_{n}dμ=n→∞lim ∫_{X}f_{n}dμ.$

Note that the pointwise limit $lim_{n→∞}f_{n}$ always exists and is measurable by this proposition.

Fatou’s Lemma

For each $n∈N$ let $f_{n}:X→R$ be a nonnegative measurable function. Then

$∫_{X}n→∞liminf f_{n}dμ≤n→∞liminf ∫_{X}f_{n}dμ.$

In the following proof we omit $X$ and $dμ$ for visual clarity.

Proof By definition, we have $liminf_{n→∞}f_{n}=lim_{n→∞}g_{n}$, where $g_{n}=f_{k≥n}f_{k}$. Now $(g_{n})$ is a monotonic sequence of nonnegative measurable functions. By the Monotone Convergence Theorem

$∫n→∞liminf f_{n}=n→∞lim ∫g_{n}.$For all $k≥n$ one has $g_{n}≤f_{k}$, hence $∫g_{n}≤∫f_{k}$ by the monotonicity of the integral. This implies

$∫g_{n}≤k≥nf ∫f_{k}$for all $n∈N$. In the limit $n→∞$ we obtain

$n→∞lim ∫g_{n}≤n→∞liminf ∫f_{n}$thereby completing the proof. $□$

Dominated Convergence Theorem

Let $(X,A,μ)$ be a measure space. For each $n∈N$ let $f_{n}:X→R$ (or $C$) be a measurable function. Suppose that the pointwise limit $f=lim_{n→∞}f_{n}$ exists almost everywhere. Suppose further that there exists an integrable function $g:X→R$ such that $∣f_{n}∣≤g$ almost everywhere for all $n∈N$. Then the functions $f_{n}$ and $f$ are all integrable, and

$n→∞lim ∫_{X}f_{n}dμ=∫_{X}fdμ.$

Proof TODO $□$