2
votes
2answers
146 views

Does uniform integrability plus convergence in measure imply convergence in $L^1$?

Does uniform integrability plus convergence in measure imply convergence in $L^1$? I know this holds on a probability space. Does it hold on a general measure space? I have tried googling. It ...
1
vote
2answers
94 views

$\int_0^1|f_n|^3\leq 1\Rightarrow \int_E |f_n|<\varepsilon$ when $|E|$ is small

Let $f_n\colon [0,1]\to\mathbb{R}$ be Lebesgue measurable with $$\int_0^1|f_n|^3\leq 1 \mbox{ for all } n.$$ Show that for all $\varepsilon>0$ there exists $\delta>0$ so that if $E\subset[0,1]$ ...
2
votes
1answer
213 views

exercise on uniform integrability

I cannot figure out the following exercise: Let $F$ be the family of functions $f$ on $[0,1]$, each of which is (Lebesgue) integrable over $[0,1]$ and has $\int_a^b|f|\le b-a$ for all ...
1
vote
1answer
60 views

Example for a sequence of functions in $\mathcal{L}^1[0,1]$

I am looking for an example of a sequence of functions $(g_n)$ that is in $\mathcal{L}^1[0,1]$ and U.I. so that the following three conditions are satisfied: $\forall \, n\, \, \vert g_n \vert ...
1
vote
1answer
90 views

Question about integration (related to uniform integrability)

Consider a probability space $( \Omega, \Sigma, \mu) $ (we could also consider a general measure space). Suppose $f: \Omega -> \mathbb{R}$ is integrable. Does this mean that $ \int |f| \chi(|f| ...
7
votes
2answers
198 views

Measure theory questions

i. If $1 < p < \infty$ and $E = \{f_a, a \in A\}$ set of measurable functions of $\mathbb{R}$ and $\sup_{a \in A} ||f_a||_p < \infty$, I want to show that for $ 0 < q < p$, $\lim ...
3
votes
1answer
232 views

A Question About Uniform Integrability

Proposition: Assume $E$ has finite measure. Let the sequence of functions $\{f_n\}$ be uniformly integrable over $E$. If $\{f_n\} \rightarrow f$ pointwise a.e. on $E$, then $f$ is integrable ...
1
vote
1answer
139 views

Uniform integrability for a single random variable

Let $X$ be a random variable. Are the following three equivalent? $X \in L^1$, i.e. $E |X| < \infty$. $X$ is uniformly integrable. That is, if given $\epsilon>0$, there exists $K\in[0,\infty)$ ...
5
votes
1answer
205 views

Equi-integrability of a single function: is it the same as summability?

Let $(\Omega, \mathcal{M}, \mu)$ be a measure space and let $f\ge 0$ be a measurable function on $\Omega$. Suppose that $f$ satisfies the following properties: For all $\varepsilon > 0$ there ...
2
votes
1answer
121 views

Uniform Integrability after composition

Let $\mu$ be a finite measure on $X \subseteq \mathbb{R}^n$. Consider the Uniformly Integrable family $\{ f_n(\cdot) \}_{n \in \mathbb{N}}$ of functions $f_n : X \rightarrow \mathbb{R}_{\geq 0}$. ...
0
votes
2answers
173 views

Non uniformly integrable stochastic process if we consider a larger range

This is probably a stupid question, but am I mislead if I think that as soon as a stochastic process indexed by $t$ (continuous time) is not uniformly integrable (UI) for a certain range of $t$, say ...
5
votes
0answers
1k views

Dunford-Pettis Theorem

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 ...
4
votes
1answer
382 views

Uniform Integrability

Let $\mu$ be a probability measure on $X$, so that $\int_X \mu(dx) = 1$. I have a family $\{f_i\}_{i=1}^{\infty}$ of functions $f_i: X \rightarrow \mathbb{R}_{\geq 0}$ such that $$ \displaystyle ...
8
votes
1answer
840 views

I want to understand uniform integrability in terms of Lebesgue integration

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 $\delta$ ...