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Definition：let $\mathcal F$ be a family of functions,if for each $ε＞0$,there is a $δ＞0$ such that for each $f∈\mathcal F$,if $A⊆E$ is measurable and $m(A)＜δ$,then $\int_A |f|＜ε$. We say $\mathcal F$ is uniformly integrable over $E$.

problem:let $\mathcal F$ be a family of functions,each of which is integrable over $E$ then $\mathcal F$ uniformly integrable over $E$ iff for each $ε＞0$,there is a $δ＞0$ such that for each $f∈\mathcal F$,if $A⊆E$ is measurable and $m(A)＜δ$,then $|\int_A f|＜ε$.

From left to right is easy,only use the defition of uniformly integrable and a inequality, but conversely how to do? This problem comes from Royden' real Analysis.

user158796
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1 Answers1

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I suppose your family consists of real valued measurable functions.

Suppose the second condition is given. If $m(A) <\delta$ then (for any $f \in \mathcal F$) $m(A \cap \{x: f(x) >0\})<\delta$ so $|\int_{A \cap \{x: f(x) >0\}} f |<\epsilon$. But this is same as $\int_{A \cap \{x: f(x) >0\}} |f| <\epsilon$. Similarly $\int_{A \cap \{x: f(x) \leq 0\}} |f| <\epsilon$ Add these two finish the proof. .

Kavi Rama Murthy
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