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Dunford–Pettis theorem : ウィキペディア英語版
Uniform integrability
Uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales. The definition used in measure theory is closely related to, but not identical to, the definition typically used in probability.
==Measure theoretic definition==
Textbooks on real analysis and measure theory often use the following definition.
Let (X,\mathfrak, \mu) be a positive measure space. A set \Phi\subset L^1(\mu) is called uniformly integrable if to each \epsilon>0 there corresponds a \delta>0 such that
\left| \int_E f d\mu \right| < \epsilon
whenever f \in \Phi and \mu(E)<\delta.
==Probability definition==
In the theory of probability, the following definition applies.
* A class \mathcal of random variables is called uniformly integrable (UI) if given \epsilon>0, there exists K\in[0,\infty) such that E(|X|I_)\le\epsilon\ \text \in \mathcal, where I_ is the indicator function I_ = \begin 1 &\text |X|\geq K, \\ 0 &\text |X| < K. \end.
* An alternative definition involving two clauses may be presented as follows: A class \mathcal of random variables is called uniformly integrable if:
*
* There exists a finite M such that, for every X in \mathcal, E(|X|)\leq M.
*
* For every \epsilon > 0 there exists \delta > 0 such that, for every measurable A such that P(A)\leq \delta and every X in \mathcal, E(|X|:A)\leq\epsilon.
The two probabilistic definitions are equivalent.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
ウィキペディアで「Uniform integrability」の詳細全文を読む



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