---- > [!theorem] Theorem. ([[Dominated Convergence Theorem]]) > Suppose $(X, \Sigma, \mu)$ is a [[measure|measure space]], $f:X \to [-\infty, \infty]$ a [[measurable function]], and $f_{1},f_{2},\dots$ [[measurable function|measurable functions]] $X \to [\infty, \infty]$ such that $\lim_{k \to \infty}f_{k}(x)=f(x)$ > for almost every $x \in X$. If there exists a [[measurable function]] $g:X \to [0, \infty]$ such that $\int g \, d\mu <\infty \text{ and }|f_{k}(x)| \leq g(x)$ > for every $k \in \mathbb{N}$ and almost every $x \in X$, then $\lim_{k \to \infty} \int f_{k} \, d\mu =\int f \, d\mu .$ > [!proposition] Corollary. (Bounded Convergence Theorem) > Suppose $(X, \Sigma, \mu)$ is a [[measure|measure space]] with $\mu(X)<\infty$. Suppose $f_{1},f_{2},\dots$ is a [[sequence]] of [[measurable function|measurable functions]] $X \to [-\infty, \infty]$ that [[pointwise converge|converges pointwise]] on $X$ to $f:X \to [-\infty, \infty]$. > If there exists $c \in (0, \infty)$ such that $|f_{k}(x)| \leq c$ for all $k \in \mathbb{N}$ and all $x \in X$, then $\lim_{k \to \infty}\int f_{k} \, d\mu =\int f \, d\mu .$ ^proposition The DCT proof relies on two lemmas which we extract here for independent use. > [!proposition] Proposition. (integrals on small sets are small) > Suppose $(X, \Sigma, \mu)$ is a [[measure|measure space]], $g:X \to [0, \infty]$ is a [[measurable function]], and $\int g \, d\mu<\infty$. Then for every $\varepsilon>0$, there exists $\delta>0$ such that $\int _{B}g \, d\mu<\varepsilon$ for every set $B \in \Sigma$ for which $\mu(B)<\delta$. > [!proof] > ^proof Fix $\varepsilon<0$. Let $h:X \to [0, \infty)$ be a [[simple function|simple]] [[measurable function]] such that $0 \leq h \leq g$ and $\int g \, d\mu -\int h \, d\mu < \frac{\varepsilon}{2}.$ The existence of such an $h$ follows from [[approximation by simple functions]] together with the characterization of the [[integral|Lebesgue integral]] in terms of supremums over integrals of simple functions. Define $H:=\max\{ h(x): x \in X \}$ (can do because $h$ is simple). > [!proposition] Proposition. (integrable functions live mostly on sets of finite measure) > ^proposition > [!proof]- Proof. ([[Dominated Convergence Theorem]]) > ~ ---- #### ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```