---- > [!definition] Definition. ([[uniformly integrable]]) > Let $(X, \Sigma, \mu)$ be a [[measure|measure space]]. A set $\Phi$ of integrable functions $X \to \mathbb{R}$ is called **uniformly integrable** if $\inf_{g \geq 0 \text{ integrable}} \sup_{f \in \Phi} \int _{\{ |f| > g \}} |f|\, d\mu =0.$ ^definition > [!equivalence] Equivalence for [[finite measure|finite measure spaces]]. > When $\mu(X)<\infty$, we have that $\Phi$ is uniformly integrable if and only if > 1. $\sup_{f \in \Phi} \|f\|_{1}<\infty$. > 2. For all $\varepsilon>0$, there exists $\delta>0$ such that $\int _{E} |f| \, d\mu <\varepsilon \text{ for all } f \in \Phi,$ for any $E \in \Sigma$ satisfying $\mu(E)<\delta$. > We also have that $\Phi$ is UI if and only if $\lim_{K \to \infty} \sup_{f \in \Phi} \int |f| \, \boldsymbol 1_{\{ |f|>K \}} \, d\mu =0.$ > [!proof] > > > $\to$. Suppose $\Phi$ UI. This direction is TODO (use [[Chebyshev's Inequality]]?). > > > $\leftarrow$. Fix $\varepsilon>0$. Have to show $\inf_{g \geq 0 \text{ integrable}} \sup_{f \in \Phi} \int _{\{ |f| > g \}} f\, d\mu < \varepsilon,$ i.e., that there exists $g \geq 0$ integrable such that $\int _{\{ |f|>g \}} f\, d\mu<\varepsilon$ for all $f \in \Phi$. We know we can obtain $\delta>0$ such that $\int _{E} |f| \, d\mu<\varepsilon$ for all $f \in \Phi$ whenever $E \in \Sigma$ satisfies $\mu(E)<\delta$. Thus we are done if we find $g$ for which $\mu(\{ |f|>g \})<\delta$ for all $f \in \Phi$. But this is immediate: we're give $S=\sup_{f \in \Phi} \|f\|_{1}<\infty$, so just take $g \equiv S$. > [!equivalence] Equivalence for [[probability|probability spaces]]. > In probabilistic language: the collection $\Phi$ is uniformly integrable if and only if $\lim_{K \to \infty} \sup_{f \in \Phi} \mathbb{E}\big[\, | f| \, \boldsymbol 1_{\{ |f|>K \}}\big]=0.$ That is, if and only if for all $\varepsilon>0$ there exists $K>0$ such that $\mathbb{E}[|f| \boldsymbol 1_{\{ |f|>K \}}]<\varepsilon$ for all $f \in \Phi$. ^equivalence ---- #### ---- #### 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 > ```