0
votes
1answer
40 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 ...
1
vote
2answers
20 views

Measures: Sequential Continuity

Disclaimer: This thread is meant as record and written in Q&A style. Let $\Omega$ be a finite measure space $\mu(\Omega)<\infty$. It is well known that a measure is continuous from above as ...
1
vote
2answers
72 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 ...
1
vote
0answers
15 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
128 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
60 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
47 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
29 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
47 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
48 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
67 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
46 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
28 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
31 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
38 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
59 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
42 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
28 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
19 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
20 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
32 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
46 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
43 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
70 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
28 views

Uniform convergence almost everywhere characterization

I am being faced with a non-trivial characterization of uniform convergence almost everywhere. Suppose $(X,S,\mu)$ is a non-necessarily finite measure space. Take a sequence of measurable functions, ...
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 ...
2
votes
1answer
96 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
75 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
98 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
43 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
49 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
32 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
62 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
104 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
31 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 ...
4
votes
0answers
225 views

Weak and Probability convergences

I have a question about this book, "Topics in Random Matrices Theory" of Terence Tao. He claims that if $$\int_{\mathbb{R}}\varphi \ ...
2
votes
0answers
74 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
61 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
49 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
50 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
26 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
45 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 ...