1
vote
1answer
23 views

Is this set measurable? (Set of points where a sequence converges)

Let $M$ be a manifold. Suppose that $u_n:M \to \mathbb{R}$ are measurable and we have $u_n(s) \to u$ a.e. in $M$. Does it follow that the set $A=\{s \in M : u_n(s) \to u(s)\}$ and $A^c$ are ...
3
votes
1answer
48 views

Convergence in measure of integrable functions implies limit is integrable?

I'm going through my handwritten notes for my upcoming exam (so not homework) and the above was stated but not proven in class. The full statement is a little different, but the above part is the only ...
0
votes
1answer
24 views

weak convergence implies point-wise convergence?

If we have a bounded sequence $\{f_n\} \in L^p[a,b]$ that converges weakly to $f$ does this mean that the converges is also pointwise?? thank you.
0
votes
1answer
24 views

If for a.e $t \in (0,T)$, $u_n(t,x) \to u(t,x)$ for a.e. $x \in \Omega$,is $u_n \to u$ a.e. in $(0,T)\times\Omega$?

If for almost all $t\in (0,T)$, we have $$u_n(t,x) \to u(t,x) \quad\text{a.e. $x \in \Omega$}$$ does this mean that $$u_n \to u \quad\text{a.e. in $(0,T)\times\Omega$}?$$ Here $\Omega$ is an open ...
3
votes
0answers
167 views
+100

Weak and Probability convergences

I have a question about this page, from Topics in Random Matrices Theory, of Terence Tao. He claims that if $$\int_{\mathbb{R}}\varphi \ ...
2
votes
0answers
68 views

Convergence of a sequence pointwise a.e. on a manifold given that it converges on a reference manifold

Let $\Gamma_t$ be a compact hypersurface for each $t \in [0,T]$. Let $Q=\Gamma_0\times(0,T)$ and $$Q_T :=\bigcup_{t \in (0,T)} \Gamma_t \times \{t\}.$$ For each $t \in [0,T]$, suppose that ...
2
votes
1answer
45 views

Variant of dominated convergence theorem

There are several variants of dominated convergence theorem. The standard one requires $f_n \to f$ a.e. and $|f_n|\leq g$ a.e. where $g$ is integrable. It can be weakened to only convergent in ...
0
votes
2answers
42 views

A function is a.e. equal to a polynomial.

Let $f\in{L^p}$. For $t>0$, let $P_{t,n}(x)$ be a collection of polynomials of degree less then or equal to $n$ in the variable $x$ and the family is given by $t$ such that ...
0
votes
1answer
23 views

weak-*-convergence of measures ==> convergence of the total mass?

Let $X = [0,1]$. Let $\mu_n$ be a sequence of regular signed Borel measures on $X$, which converges to a measure $\mu$ on $X$ in weak-star, i.e. for any $f\in C_0(X)$, we have $\int_X f \mu_n(dx) \to ...
1
vote
1answer
24 views

Why is this set an event?

As a part of a setup for another problem, my text remarks that it can be used without a proof that if $X_1, X_2, \ldots$ are random variables then $$C:=\{ \omega\in\Omega \ | \ \sum X_n(\omega) \ ...
0
votes
0answers
32 views

Properties convergence of random variables

I want to understand the concepts of convergence of random variables better and therefore I wanted to find out how certain concepts that are straightforward for canonical convergence behave in this ...
2
votes
1answer
25 views

Why is $s-\lim_{n\to\infty} f_n = f\Longleftrightarrow~\forall g\in L_{\mu}^{\infty}: \lim_{n\to\infty} f_n g\, d\mu=\int fg\, d\mu$?

We had some different definitions concerning convergence. And I am a bit confused about that. First of all I give you some preparing definitions and then the definitions I mean. Let $E$ be a Banach ...
3
votes
1answer
40 views

Convergence almost everywhere implies convergence in measure, the proof thereof

Let $(E, \mathcal{E}, \mu)$ be a measure space, and $(f_n)_{n\in\mathbb{N}}$ and $f$ be measurable functions $(E, \mathcal{E}, \mu)\longrightarrow (\mathbb{R}, \mathcal{B})$. The first part of Theorem ...
1
vote
1answer
22 views

An unknown function

I run into a function: $1_{[-n, n]^r}$. I guess this function equals 1 whenever x falls into $[-n, n]^r$. Am I right? I met this function in an analysis paper which deals with measure and density of ...
3
votes
1answer
74 views

Why is $g_n:=\inf\{f_n,f_{n+1},…\}$ integrable if $f_n$ are?

This question is motivated by the proof of the Fatou's lemma. My text defines $g_n:=\inf\{f_n,f_{n+1},\ldots\}$ and states that it's Lebesgue integrable (each $f_n$ is). We proved that point-wise ...
1
vote
0answers
38 views

Convergence of $L^1$ functions

Given that $\Omega$ is bounded and $a_{ij}(u_{k}) \rightarrow a_{ij}(u)$ in $L^{1}(\Omega)$, $a_{i0}(u_{k}) \rightarrow a_{i0}(u)$ in $L^{1}(\Omega)$, $\frac{\partial u_{k}}{\partial x_{j}} ...
-1
votes
1answer
27 views

Equivalence of these three statements about convergence of events in a $\sigma$-algebra

Could you explain to me why these three statements are equivalent. Everything takes place in a $\sigma$-algebra $\mathcal A$. Let $\{A_n\}$ be a sequence of events in $\mathcal A$. $1_A$ denotes the ...
1
vote
1answer
37 views

Convergence in $L^p$ of $f_x(y)=f(y-x)$

Looking through practice problems to get ready for my exam, I found the following one which confuses me a bit: Let $1\le p<\infty$ and $f\in L^p(\mathbb{R})$. For $x\in\mathbb{R}$ let ...
1
vote
1answer
36 views

If $(f_n)$ is Cauchy in the $L^2$-norm, then is $(f'_n)$ Cauchy in the $L^2$-norm?

Let $(f_n)$ be a sequence in $H^1(a,b)=\{f\in L^2(a,b);\;f'\in L^2(a,b)\}$, where $-\infty<a<b<+\infty$. If $(f_n)$ is a Cauchy sequence in the norm $\|\cdot\|_{L^2}$, is it possible to ...
0
votes
0answers
32 views

Why the convergence of the following operator consequence is strong?

Given a consequence of the following functional operators in $\mathcal{B}(L_{p}[0,1])$, $p \in [1,\infty)$ $$ (A_{n}x)(t) = \sum_{k=1}^{n} n \int_{t_{k-1}}^{t_{k}} x(s)ds \chi_{k,n}(t), $$ where ...
1
vote
1answer
30 views

Relation between convergences in $L^{p}$ for probability spaces.

I have read that for a probability space $(\Omega,\Sigma,P)$ it is true that $f \in L^{p}(\Omega,\Sigma,P)$ implies $f \in L^{q}(\Omega,\Sigma,P)$ if $p>q$, and hence $L^{2} \subset L^{1}$. I'm ...
-1
votes
1answer
29 views

The convergence in $L^{p_1}$ and $L^{p_2}$

Suppose $f_k$ is a sequence of $M-$measurable function. Let $p_1$ and $p_2\in[1,\infty)$,and $f_k\in L^{p_1}\cap L^{p_2}$.Also suppose $\exists g\in L^{p_1}$ and $h\in L^{p_2}$ s.t. $f_k\rightarrow g$ ...
1
vote
1answer
45 views

Convergence in $L^1$ coincides with convergence a.e.

On the measure space $(X,F,m)$, if $\lim\limits_{n\rightarrow \infty} \int\limits_X |f_n-f|\mathrm{d}m=0$ and $\lim\limits_{n\rightarrow \infty} f_n =g$ almost everywhere, then prove that $f=g$ ...
1
vote
1answer
73 views

Almost sure weak convergence of empirical measure

Do empirical measures converge weakly to the measure almost surely? In particular suppose $\mu$ is a Borel probability measure on $\mathbb R^d$ and that $X_1,X_2,\dots$ are IID drawn from $\mu$. Let ...
2
votes
2answers
118 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 ...
2
votes
1answer
68 views

Show that $f_n1_{A_n}$ convergences in mean

Consider the measurable space $(\Omega,\mathcal{A},\mu)$. Let $f,f_1,f_2,\ldots$ be measurable functions on that measurable space and $A,A_1,A_2,\ldots\in\mathcal{A}$. Let $(f_n)$ converge in ...
1
vote
1answer
46 views

Stochastic convergence (product)

Consider the measurable space $(\Omega,\mathcal{A},\mu)$. Let $f,f_1,f_2,\ldots$ be measurable functions and $A,A_1,A_2,\ldots\in\mathcal{A}$. Let $(f_n)_{n\in\mathbb{N}}$ be convergent stochastically ...
0
votes
2answers
84 views

Does $\sum_{m=1}^{\infty}f_{m}\left(x\right) $ converge for almost every $x$ in $X$?

Let {$f_{m}$} be a sequence of measureable real-valued functions in $\left(X,\mathrm{\mathcal{M}},\mu\right)$. Suppose ${\displaystyle \sum_{m=1}^{\infty}\left\{ {\displaystyle ...
1
vote
1answer
56 views

convergence of convolutions and approximation of unity

Let $\phi : \mathbb{R}\rightarrow \mathbb{R}$ be an integrable function with $\int \phi(x)dx = 1$. Let us define $\phi_\delta = \delta^{−1}\phi(\delta^{-1}x)$. Show that for every continuous ...
1
vote
0answers
63 views

Dominated Convergence on risk measures

This is a quite specific question and I am not able to provide the whole background (e.g. what a risk measure is). If someone knows that would be great. I am having difficulties understanding a ...
1
vote
2answers
46 views

Example of an integral not converging

Consider a measurable space $(\Omega,\mathcal{A},\mu)$ with $\mu(\Omega)<\infty$. Let $f_1,f_2,\ldots$ be bounded measurable functions so, that $f_n\to f$ uniformly. Then $f$ is measurable ...
1
vote
1answer
45 views

Convergence of $\int_{0}^{2}{k^{p}x^{k} \over 1 + x^{2k}}\,{\rm d}x\quad\mbox{with}\quad k = 1,2,3,\ldots\quad\mbox{and constant}\quad p$

Consider the integrals $$ \int_{0}^{2}{k^{p}x^{k} \over 1 + x^{2k}}\,{\rm d}x\quad\mbox{with}\quad k = 1,2,3,\ldots\quad\mbox{and constant}\quad p. $$ For what values of $p$ do the integrands have an ...
2
votes
1answer
86 views

Do The Integrals Tend to 0?

Consider the integrals $\int_1^\infty \frac{k}{x^2+k^p\cos^2x}dm(x),$ where $m$ is the Lebesgue measure. For what $p$ do the integrands have an integrable majorant? For what $p$ do the integrals tend ...
1
vote
0answers
60 views

Convergence of the Integrals

Consider the integrals $\int_1^\infty\frac{x^2+kx}{x^4+k^px^2+k^2}dm(x)$ for k=1,2,3,.... For what p do the integrands have an integrable majorant? For what p do the integrals tend to $0$? I'm ...
2
votes
1answer
68 views

Help with Homework Problem

Let $f_k(x)=|x-1/k|^{-2/3} (k=1,2,3,...).$ Do the $f_k$ have an integrable majorant, meaning a function bounding $f_k$ that satisfies the dominated convergence theorem, on the interval [0,1] with ...
0
votes
1answer
54 views

Another Question on Convergence

Let $f_n$ be a sequence of non-negative measurable functions on $(X,\mathscr{A},\mu)$ such that for every $x\in X$, the sequence $f_n(x)$ is convex. Show that the limit of $f_n(x)$ as n goes to ...
0
votes
1answer
32 views

Convergence Problem With Parametric Family

Let $f_t(x)=\left|\sin tx\right|^t$, $x\in[0,1]$, $t\in(0,+\infty)$. Does the parametric family converge on $[0,1]$ in measure with respect to the Lebesgue measure as $t$ approaches $+\infty$? Does it ...
1
vote
0answers
41 views

Infinite discounted sum of weakly dependent Normal random variables

Say I have the expected value of a sum of weakly dependent Normal random variables of the form $\mathbb{E}\left[\sum_{n=1}^\infty a^n X_n\right]$, where $0<a<1$. I was wondering under what ...
1
vote
1answer
37 views

Does the following function converge to zero almost certainly?

Let $X_n(w) = nI_{(n-1,n)}(w)$ be a function on the real numbers with the Borel sigma algebra. Consider Lebesgue measure, denoted by $\mu$. Can this be a random variable? Does this converge to 0 ...
0
votes
1answer
39 views

Showing $\mathbb{E}(\max_{k \leq n}|X_k|)/\sqrt{n} \to 0$ as $n \to \infty$.

If $(X_n: n \in \mathbb{N})$ is an identically distributed sequence in $\mathbb L^2(\mathbb{P})$, I want to show that $$\displaystyle\frac{\mathbb{E}\left(\max\limits_{k \leq ...
0
votes
0answers
44 views

Weak convergence implying a.s convergence of a subsequence

Given $(f_n)$ a sequence in $L^{\infty}(\mu)$ which weakly converges to $f\in L^{\infty}(\mu),$ i.e. convergence in $\sigma(L^{\infty}, L^1),$ where $\mu$ is a probability measure. Is it true that we ...
1
vote
0answers
42 views

Necessary Sufficient Conditions for Almost sure convergence

$X_n$ is an i.i.d. sequence and let $S_n=\sum_{i=1}^nX_i$. (a) Prove that $(S_n-C_n)/n\rightarrow0$ almost surely for some real sequence $\{C_n\}$ if and only $E(|X_1|)<\infty$ and in that case ...
0
votes
0answers
65 views

Parameter integral and continuity (Theorem of Lebesgue)

I already kept myself busy with a proof concerning the Theorem of Lebesgue and differentiation of a parameter integral. Unfortunately I did not get an answer there yet. Now my task is nearly the ...
3
votes
1answer
71 views

Uniform convergence and convergence of integrals

Question: $(X,\mathcal{M},\mu)$ measure space. Suppose that $\{f_n\}\subset L^1$ and $f_n\rightarrow f$ uniformly. Show that if $\mu(X) <\infty$, then $\int f_n\rightarrow \int f$. Proof: I ...
1
vote
1answer
42 views

convergence in probability is equivalent to convergence in this metric

Let $\mathcal{L}_0(\Omega,\mathcal{A},P)$ be the space of all real random variables on a probability space $(\Omega,\mathcal{A},P)$. Then I have showed that $\rho(X,Y)=E[|X-Y|/(1+|X-Y|)]$ is a metric ...
0
votes
0answers
65 views

Theorem of Lebesgue and differentiation of a parameter integral

Let $(a,b)\subset\mathbb{R}$ be an interval and let $\left\{f_t\colon\Omega\to\mathbb{R}\right\}_{t\in (a,b)}$ be a family of measurable functions on the measurable space ...
0
votes
1answer
54 views

when convergence in measure implies convergence almost surely

If I have a discrete measure space $(\Omega,\mathcal{A},\mu)$, that is when $\Omega$ is countable and $\mathcal{A}$ is the $\sigma$-field over $\Omega$ containing all the subsets of $\Omega$, then ...
3
votes
2answers
111 views

When does almost everywhere convergence imply convergence in measure?

Let $f_n$ be a sequence of measurable functions on a finite measure space. Is it true that If every subsequence of $\{f_n\}$ has a subsequence which converge to $f$ almost everywhere, then $f_n$ ...
2
votes
1answer
122 views

Lebesgue's Dominated Convergence Theorem questions

Assume that $f_n \to f$ almost everywhere, with $f_n$ integrable for all $n$ and $g$ is an integrable function such that $\lvert f_n \rvert \le g$. (A) Then $f$ is integrable and - $$\int f\,\mathrm ...
1
vote
0answers
200 views

Clarification of “Convergence almost everywhere implies convergence in measure”

I have looked over the proofs that show convergence a.e. imply convergence in measure. I understand the proofs, but I do not understand why one must go into such detail. It seems as though one ...