0
votes
1answer
12 views

a bounded function is converge in measure, then its limit is also converge

If a series function, ${f_n} \rightarrow f $ in measure $\mu$, and $|f_n| \leq M$, how to show that $|f| \leq M$? My instructor gave a hint as follows, but I do not believe the first inequality ...
0
votes
1answer
20 views

Approximation in $L^2(\Omega)$

I want to prove that if $f_n\to f$ in $L^2(R)$ then $f_n(X)\to f(X)$ in $L^2(\Omega)$ for each random variable X. I think of using the dominated convergence theorem, having the puntual convergence, ...
1
vote
1answer
31 views

proof of DCT with weak condition(almost everywhere)

I have a question about a proof of the dominating convergence theorem, with weak requirements. Before I show the proof from the book, note that in my book you are allowed to integrate functions that ...
0
votes
1answer
26 views

dense subspace of $L^2(\Omega\times(0,T))$

I am trying to prove that the functions $f(\omega,t)=g(\omega)h(t)$ where $g\in G,\: h\in H,$ are dense in $L^2(\Omega\times(0,T))$ if $G$ is dense in $L^2(\Omega)$ and $H$ is dense in $L^2((0,T))$. ...
0
votes
1answer
18 views

Example about Dominated Convergence Theorem

So I was reading my textbook about Dominated Convergence Theorem: I have $(X,\mathscr{F},\mu)$ as a measure space I have $f,f_n,: X\to [-\infty, \infty], g:X\to [0,\infty]$ integrable and it is the ...
0
votes
1answer
41 views

$P(X_n < a$ i.o. and $X_n > b$ i.o.$) = 0$ for all $a < b$ implies that $lim_{n \rightarrow \infty} X_n$ exists a.e.

Suppose for $a<b$ we have $P(X_n < a$ i.o. and $X_n > b$ i.o.$) = 0$. Then $lim_{n \rightarrow \infty} X_n$ exists a.e. but may be infinite. Here "i.o." means "infinitely often"; for any ...
0
votes
1answer
19 views

Essential Uniform Convergence Implication

Greetings Mathematics Community. I believe that I am thinking too hard about the following problem and would like some guidance in solving it. Let $X$ have finite measure and let $f_n:X \to ...
4
votes
2answers
35 views

Cauchy convergence in probability implies the existence of a (finite a.e.) limit X

Cauchy convergence of a sequence $X_n$ of random variables in probability implies the existence of an X (finite a.e.), such that $X_n$ converges to X in probability. The problem's hint suggests ...
0
votes
0answers
26 views

Almost sure downward convergence with some conditions implies convergence in $L^1$

If $X_n\downarrow X$ a.s., each $X_n$ is integrable and $inf_n E[X_n] > -\infty$, then $X_n \rightarrow X$ in $L^1$. As far as I know, "$X_n\downarrow X$ a.s." means that for every n, $X_n ...
1
vote
2answers
44 views

Does $X_n \xrightarrow{\mathbb P} c \in \mathbb R$ imply $\phi(X_n) \xrightarrow{\mathbb P} \phi(c)$ in this case?

Let $X_n \xrightarrow{\mathbb P} c \in \mathbb R$ and $\phi: \mathbb R \to \mathbb R$ be bounded, continuous in $c$, and $\phi(c)=0$. Show that $\mathbb E\left[\phi(X_n)\right]\to0.$ I was going ...
0
votes
1answer
34 views

Sequence of functions that converges a.e. but not in the $L^1$ norm [closed]

How can I construct a sequence of functions that converges a.e. but it does not in the $L^1$ norm?
0
votes
1answer
43 views

Visual understanding of convergence of domains in the sense of Fisher

In these lecture notes by Ueltschi here, I found in Definition 2.3 a peculiar type of convergence. Especially the second property is hard for me to visualize what it means, could anybody try to ...
3
votes
2answers
35 views

Measures: Sequential Continuity

Disclaimer: This thread is meant as record and written in Q&A style. Let $\Omega$ be a measure space. It is well known that a measure is continuous from below as well as from above: ...
1
vote
2answers
76 views

The limit of $\int_{0}^{1}\frac{\sqrt{n}}{1+n\ln(1+x^2)}dx$ as $n\to\infty$

The task is to calculate $$\lim_{n\to\infty}\int_{0}^{1}\frac{\sqrt{n}}{1+n\ln(1+x^2)}dx$$ I tried various estimates I know to find the dominating integrable function and nothing worked. Does anyone ...
2
votes
0answers
16 views

scalar dimension to the approximation of an integrable function

Let $(M,\mathcal{A},\mu)$ space of probability. If $f\in L^1(\mu)$ then $f=\displaystyle{\lim_{n \rightarrow +\infty}\sum_{i=1}^n}\alpha_i\chi_{C_i}$ where $C_i\in \mathcal{A}$ and $\alpha_i \in ...
4
votes
4answers
162 views

Convergence in measure implies pointwise convergence?

In showing that we can replace pointwise convergence with convergence in measure in the Lebesgue Dominated Convergence Theorem, I made the following claim: 1.) $f_n\to f$ in measure ...
1
vote
2answers
61 views

Two questions about convergence in measure

I am currently studying for my analysis comprehensive exam and have a few questions about convergence in measure. First of all I know that for a sequence $\{f_n\} \in L^p(E)$, $1 \le p < \infty$, ...
2
votes
1answer
48 views

Convergence in measure and convergence in $L^p$

If $f_n$ is convergent to $f$ in measure and $\|f_{n}(x)\|_{L^{p}(\mathbb{R})}=\|f(x)\|_{L^{p}(\mathbb{R})}$. Does it implies that $f_n$ is convergent in $L^p$?
0
votes
1answer
31 views

Criteria for measure convergence implying convergence a.e.

Suppose the function $g_n = \sup_{m \geq n} |f_n-f_m|\to 0$ in measure. Show $f_n \to f$ a.e. Suppose instead that $\sum_{n=1}^{\infty} m\{ |f_n - f|>\epsilon\} < \infty$. Show $f_n \to f$ a.e. ...
0
votes
1answer
31 views

Sequence of sets $f_n$ that converge almost everywhere to $f$ but not almost uniformly to $f$?

I've been trying to think of a simple example of a sequence of sets $f_n$ that converge almost everywhere to $f$ but not almost uniformly to $f$. Suppose $f$ is the zero function for the sake of ...
1
vote
0answers
48 views

Measurable set of points where a measurable sequence fails to converge

Let $\{f_n\}$ be a sequence of measurable functions. Prove that the set of points $x$ such that $\{f_n(x)\}$ fails to converge as $n\to\infty$ is measurable. My first attempt was Suffices to show ...
2
votes
1answer
49 views

$f_n(x) = x(1-x)^n$ converges uniformly to $0$ on $[0, 1]$?

I am trying to show that the sequence of functions $f_n(x) = x(1-x)^n$ converges uniformly to $0$ on $[0, 1]$. Well at $0$ and $1$, $f_n(x) = 0$ for all $n$. So let $x \in (0, 1)$. $f_n(x)$ ...
1
vote
1answer
35 views

$F_n \overset{w}{\to} F$, and $F$ is continous. Show that $F_n$ converges to $F$ uniformly on $\mathbb{R}$

$\{F_n\}$ and $F$ are distribution functions, and $F$ is continuous on $\mathbb{R}$. If $F_n$ converges weakly to $F$, show that $$ \sup_x | F_n(x) - F(x) | \to 0, n \to \infty $$ I know that ...
1
vote
3answers
90 views

Set of points at which sequence of measurable functions converge (another approach)

Question is to prove that : Set of all points at which a sequence of measurable functions converge is a measurable set.. What i have tried is as follows : We are looking at the following set : ...
1
vote
1answer
49 views

Space of Gaussian Functions is Closed in $L^2$

Let $\Omega, \mu$ be a probability space. A measurable function $f: \Omega \rightarrow \mathbb{R}$ is called Gaussian if $$\mu (f^{-1}(A))=\frac{1}{\sigma \sqrt{2\pi }}\int_Ae^{-x^2/2\sigma ^2} dx$$ ...
0
votes
1answer
31 views

product measure of weak (-*-) converging signed measure is weak-(*)-converging?

Assume that we have a sequence of signed measures $\mu_n$ on $[0,1]$ that converge weak(-*) to $\mu$, that means: For all continuous and bounded functions $f:[0,1] \rightarrow R$ we have $\int_0^1 ...
3
votes
1answer
53 views

extending the convergence of measures

Let $F$ the real vector space of all applications $\phi: X \times Y \rightarrow \mathbb{R}$ where $(X,\mathcal{B}_1, \mu)$, $(Y,\mathcal{B}_2 )$ measurable spaces with $X$ and $Y$ are compact metric ...
3
votes
1answer
47 views

Convergence in total variation

There are the very basic convergence types in probability theory: almost sure, in $L^p$-norm, in measure and in distribution. Besides that there is the concept of convergence in total variation norm. ...
0
votes
0answers
41 views

Limit of measurable function is measurable

This question has been asked already here but I didn't get a satisfactory solution and didn't want to bring up an old question. Here is the question : Let $\{f_n\}$ be a sequence of measurable ...
2
votes
2answers
61 views

Properties of a sequence of iid rv's

I cannot do part a), and Im fairly sure that b) and c) will follow from it. If possible could I please have a solution to part a) and hints if you feel necessary to parts b) and c).
2
votes
1answer
22 views

Modes of Convergence of a particular random variable

Let $X_n \sim U([-1/n,1/n])$ be uniform random variables on $[-1/n,1/n]$ for $n \in \mathbb{N}$. Do the $X_n$ converge, and if yes in what sense? I think it converges pointwise as for any $x \in ...
1
vote
1answer
45 views

Are the measurable subsets closed under the symmetric distance metric?

Define the following pseudo-metric on the set of measurable subsets of $R$: $$D(A,B)=\operatorname{Length}((A\setminus B) \cup (B\setminus A)),$$ i.e., the distance between $A$ and $B$ is ...
3
votes
0answers
31 views

Asymptotic Bounds for the distribution of $f_n(X_n)$.

Let $\{X_n\}_{n \in \mathbb{N}}$ be a sequence of $\mathbb{R}^{k}$-valued random variables defined on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ converging almost surely to $X$. ...
2
votes
1answer
22 views

Convergence in distribution problem

I want to prove that, in $(\mathbb{R},B(\mathbb{R}))$, we have that $\frac{1}{n}\sum_{i=1}^{n}\delta_{\frac{i}{n}}$ converges to $U_{[0,1]}$. We need to prove, by definition, that $\lim_{n \to ...
1
vote
1answer
27 views

convergence in measure implies the composition of the sequence of functions and a continuous function also converges in measure

Let $D$ be a measureble set in $\mathbb{R}^n$. Suppose $\mu(D)<\infty$. Let $\phi: D\times \mathbb{R}\to \mathbb{R}$ be a continuous function such that for almost every $x\in D$, ...
0
votes
1answer
33 views

Show that $\mu(f_n^+) \rightarrow \mu(f^+) $ and $\mu(f_n^-) \rightarrow \mu(f^-) $, using Fatou's Lemma.

I'm starting learning about Fatou's lemma. How would you apply it to solve the following problem: Let $g^+ = max (g,0)$ and $g^- = max (-g,0)$. Let $f_n$ be integrable on measure space with measure ...
1
vote
1answer
58 views

Counterexamples for L1-convergence not implying L2-convergence and vice versa

I am trying to find counterexamples for the following statements. Let $\{f_n\}$ be a sequence in $L^1(\mathbb{R}^d) \cap L^2(\mathbb{R}^d)$, and let $f$ also be in $L^1(\mathbb{R}^d) \cap ...
2
votes
1answer
49 views

limit of the integrations of a sequence of integrable functions

Let $(f_n)^\infty_{n=1}$ be a sequence of Lebesgue integrable functions on $[0,1]$ such that $f_n$ converges to $f$ almost everywhere in $[0,1]$. Suppose further (a). ...
3
votes
1answer
73 views

can $L^p$ norm convergence and pointwise monotonic imply pointwise convergence?

Let $(f_n)_{n=1}^\infty$ be a sequence of measurable function such that $\lim_{n\to\infty}||f_n-f||_p=0$. If for any $x\in \Omega$, $\{f_{n}(x)\}_{n=1}^\infty$ is a monotonic sequence, can we deduce ...
1
vote
0answers
31 views

Does convergence of integrals imply a.e. convergence of functions?

Let $\{u_n\}_{n\in\mathbb{N}}\subseteq L^{\infty}([0,T],U)$ be a sequence of measurable functions, where $U=[0,\overline u]$ is a compact set, and suppose that $\lambda(t,du)$ is a probability kernel ...
3
votes
1answer
106 views

Pointwise a.e. convergence and weak convergence in Lp

I'm trying to prove the following theorem: Let $\{f_n\}\subset L^p(\Omega)$, $f_n \rightharpoonup f$ in $L^p(\Omega)$ ($\Omega\subset\mathbb{R}^n$ is open and bounded, $1\leq p \leq \infty$) and $f_n ...
4
votes
1answer
58 views

Dense subspace of the space of measures on the torus $\mathbb{T}$.

Every measure $\mu$ on the torus $\mathbb{T}$ is the weak-$\ast$ limit of a sequence of absolutely continuous measures on $\mathbb{T}$ with $C^{\infty}$ densities. I'd like to see a proof of this ...
2
votes
1answer
91 views

Exercise on Dominated convergence theorem

Consider the sequence $f_n=(-1)^n \frac{x}{\log(1+x)} \chi_{(0,1/n)}(x)$. Is it true that $$ \sum_n \int_X f_n d\mu= \int_X \sum_n f_n d\mu$$ with $ X=(0,1)$? I was thinking about using the ...
0
votes
1answer
51 views

Does $L^p$ convergence imply convergence of integrals?

If $L^p-\lim_{t\rightarrow\infty} f_t = f$ ($p > 1$), is it the case that $\lim_{t\rightarrow\infty}\int f_t^p = \int f^p$?
0
votes
1answer
35 views

Converse of existing question on L^p convergence

My question is about this: Convergence in $L^p$ of $f_n$ implies convergence in $L^1$ of $|f_n|^p$ and $f_n^p$ It was shown that the author's question was indeed true by the use of MVT. Is the ...
3
votes
1answer
99 views

Convergence of characteristic functions on hypercube

I have a question regarding the following partition of a hypercube $H_{R}(x)$ centered at $x$ with sides of length $R$ in $\mathbb{R}^{n}$: Consider this hypercube $O = H_{R}(x) = ...
2
votes
2answers
69 views

Convergence of sums using Hilbert space techniques [duplicate]

Let $a_n$ be a sequence of real numbers such that $\sum_{n=1}^{\infty} a_nb_n < \infty$ for any sequence $b_n$ satisfying $\sum_{n=1}^{\infty}b_n^2 < \infty$. Prove that ...
3
votes
0answers
44 views

Problem of convergence of characteristic functions

Let $\Omega \subset \mathbb{R}^{n}$ be bounded and open. Assume that $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a Caratheodory function which means $a(x,.,.)$ is ...
4
votes
0answers
50 views

Strongly convergence in L^2

Let the sequence $(f_n(x,u))$ such as, for all $n,$ $f_n$ is Caratheodory, and $|f_n|\leq g$ where $g \in L^1(\Omega)$ Let $u_n \in H^1_0$ such as it is strongly convergent to $u$ in $L^2$ and a.e ...
1
vote
1answer
33 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 ...