What is uniform integrability?

Asked Feb 14 '15 at 21:19

Active Feb 14 '15 at 21:56

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I know two definitions of uniform integrability:

(Rudin - Real and Complex Analysis). Let $(X,\mathfrak{M},\mu)$ be a positive measure space. A set $\Phi\subset L^1(\mu)$ is called uniformly integrable if to each $\epsilon>0$ corresponds a $\delta>0$ such that $$\left|\int_E fd\mu\right|<\epsilon\tag{1}$$ whenever $f\in\Phi$ and $\mu(E)<\delta$.
(Royden & Fitzpatrick - Real Analysis) Same thing except that $(1)$ is replaced by $$\int_E|f|d\mu<\epsilon.$$

Are the two equivalent? Clearly 2. implies 1., but I am not sure whether or not 1. implies 2.

The second one a lot more convenient to use in proofs, but sometime I don't know if "uniformly integrable" only assumes the first one.

asked Feb 14 '15 at 21:19

Peter

Try to prove 1 implies 2. – GEdgar Feb 14 '15 at 21:34
As a technical side note, Royden actually defines uniform integrability for a collection of measurable functions instead of integrable ones (see this answer of mine), though he gives this equivalence as an exercise (5.6.50, p. 96). – Alp Uzman Nov 10 '16 at 23:19

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