equivalent conditions for uniform integrability

Let $(\Omega,\mathcal{F},\mu)$ be a measure space and $S$ be a bounded subset of $L^{1}(\Omega,\mathcal{F},\mu)$ . That is, $\int|f|\,d\mu$ is bounded over $f\in S$ . Then, the following are equivalent.

1.

For every $\epsilon>0$ there is a $\delta>0$ so that

$\int_{A}|f|\,d\mu<\epsilon$

for all $f\in S$ and $\mu(A)<\delta$ .
2.

For every $\epsilon>0$ there is a $K>0$ satisfying

$\int_{|f|>K}|f|\,d\mu<\epsilon$

for all $f\in S$ .
3.

There is a measurable function $\Phi\colon\mathbb{R}\rightarrow[0,\infty)$ such that $\Phi(x)/|x|\rightarrow\infty$ as $|x|\rightarrow\infty$ and

$\int\Phi(f)\,d\mu$

is bounded over all $f\in S$ . Moreover, the function $\Phi$ can always be chosen to be symmetric and convex.

So, for bounded subsets of $L^{1}$ , either of the above properties can be used to define uniform integrability. Conversely, when the measure space is finite, then conditions (2) and (3) are easily shown to imply that $S$ is bounded in $L^{1}$ .

To show the equivalence of these statements, let us suppose that $\int|f|\,d\mu<L$ for $f\in S$ .

(1) implies (2)

For $\epsilon>0$ , property (1) gives a $\delta>0$ so that $\int_{A}|f|\,d\mu<\epsilon$ whenever $f\in S$ and $\mu(A)<\delta$ . Choosing $K>L/\delta$ , Markov’s inequality gives

\mu(|f|>K)\leq K^{-1}\int|f|\,d\mu\leq L/K<\delta

and, therefore, $\int_{|f|>K}|f|\,d\mu<\epsilon$ .

(2) implies (3)

For each $n=1,2,\ldots$ , property (2) gives a $K_{n}$ satisfying

\int(|f|-K_{n})_{+}\,d\mu\leq\int_{|f|>K_{n}}|f|\,d\mu\leq 2^{-n}.

Without loss of generality, the $K_{n}$ can be chosen to be increasing to infinity, so we can define $\Phi(x)=\sum_{n}(|x|-K_{n})_{+}$ . Then,

\int\Phi(f)\,d\mu=\sum_{n}\int(|f|-K_{n})_{+}\,d\mu\leq\sum_{n}2^{-n}=1.

(3) implies (1)

First, suppose that $\int\Phi(f)\,d\mu<M$ for $f\in S$ . For $\epsilon>0$ , the condition that $\Phi(x)/|x|\rightarrow\infty$ as $|x|\rightarrow\infty$ gives a $K>0$ such that $\Phi(x)/|x|\geq 2M/\epsilon$ whenever $|x|>K$ . Setting $\delta=\epsilon/2K$ ,

\begin{split}\displaystyle\int_{A}|f|\,d\mu&\displaystyle\leq\int_{|f|>K}|f|\,% d\mu+K\mu(A)\\ &\displaystyle<(\epsilon/2M)\int_{|f|>K}\Phi(f)\,d\mu+K\delta\\ &\displaystyle<\epsilon/2+\epsilon/2=\epsilon.\end{split}

whenever $\mu(A)<\delta$ and $f\in S$ .

Title	equivalent conditions for uniform integrability
Canonical name	EquivalentConditionsForUniformIntegrability
Date of creation	2013-03-22 18:40:17
Last modified on	2013-03-22 18:40:17
Owner	gel (22282)
Last modified by	gel (22282)
Numerical id	7
Author	gel (22282)
Entry type	Theorem
Classification	msc 28A20