In his nice answer about uniform integrability, Did did show that condition (C) below is equivalent to (C1) and (C2) together. (C) For every $\varepsilon\gt0$, there exists a finite $c$ such that, for every $X$ in $\mathcal H$, $\mathrm E(|X|:|X|\geqslant c)\leqslant\varepsilon$. (C1) There exists a finite $C$ such that, for every $X$ in $\mathcal H$, […]

I think part a) can be solved by using $X_n=\frac{1}{n}\chi_{[0,n^2]}$ Not sure about part b).

The Dunford-Pettis Theorem (see Uniform Integrability Wiki) states that: A class of random variables $X_n \in L^1(\mu)$ is Uniformly Integrable if and only if it is relatively weakly compact. Now $X_n \in L^1(\mu)$ means that $\sup_n \int_\mathbb{X} |X_n(x)| \mu(dx) < \infty$. A family $\mathcal{X}$ is relatively weakly compact if for every sequence $\{X_n\}$ in $\mathcal{X}$ […]

Suppose that $X$ have finite measure, let $1<p<\infty$, and suppose that $f_n : X\rightarrow \mathbb{R}$ is a sequence of measurable functions such that $\sup_n \int_X |f_n|^p d\mu < \infty$. Show that the sequence $f_n$ is uniformly integrable. In another word, show that $$\sup_n \int_X |f_n|^p d\mu < \infty \;\;\;\Longrightarrow\;\;\; \sup_n \int_{|f_n|>M} |f_n| d\mu \rightarrow 0 […]

According to my Real Analysis textbook, a family $\scr{F}$ of measurable functions on $E$ is said to be uniformly integrable over $E$ provided for each $\epsilon$ $>$ $0$, there is a $\delta$ $>$ $0$ such that for each $f$ $\in$ $\scr{F}$, if $A$ $\subseteq$ $E$ is measurable and $m(A)$ $<$ $\delta$, then $\int_{A}$$|f|$ $<$ $\epsilon$. […]

I need to show that if $|f_n|\leq g$, and $g$ is integrable, then $\{f_n\}$ is uniformly integrable, i.e., $\underset{a\rightarrow\infty}\lim\sup_n\int_{[|f_n|\geq a]} |f_n| d\mu=0$. Here is how I thought about it: Since $|f_n|\leq g$, and $g$ is integrable, then each $f_n$ is integrable. Thus, for each $n$, $|f_n| I_{[|f_n|\geq a]}\leq |f_n|$ and $|f_n| I_{[|f_n|\geq a]}\rightarrow 0$ almost […]

Let $(X,\mathcal{M},\mu)$ be a measure space and $\{f\}$ be a sequence of functions on $X$, each of which is integrable over $X$. Show that $\{f_n\}$ is uniformly integrable if and only if for each $\varepsilon \gt 0$, there is a $\delta \gt 0$ such that for any natural number $n$ and measurable subset $E$ of […]

I’ve come across a statement that implies that non-negative martingales for which $\{M_{\tau}\mid \tau \ \rm{stopping} \ \rm{time}\}$ is not uniformly integrable exist. I personally can’t think of an example though. I’ve considered gambling strategies and Brownian Motions but none seem to work. Is there anyone who can think of an example and help me […]

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