1
vote
2answers
71 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
2answers
24 views

Limiting variable in interval: Lebesgue Dominated Convergence

So I am pretty comfortable using the LDCT for definite integrals and summations, but I am looking at a problem that has the interval as a function of the limiting variable, i.e.: $$\lim_{n\to\infty} ...
6
votes
1answer
51 views

About the $\lim_{n \to+\infty} \frac{1}{n}\int_0^1 \log(1+e^{nf(x)})\,dx$ (Rudin's exercise)

Problem (Rudin, R&CA chapter 2, no. 25) (i) Find the smallest positive constant $c$ such that $$ \log(1+e^t) \le c+t , \qquad t \in (0,+\infty). $$ (ii) Does $$ \lim_{n ...
4
votes
1answer
39 views

How to show convergence in distribution

Let $([0,1],B,\lambda)$ (B Borel Sigma-algebra) and $\lambda$ the Lebesgue measure. I want to show that this sequence converges in distribution. $$X_n(\omega)= \left(\begin{matrix} 1, & ...
1
vote
1answer
42 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 ...
1
vote
0answers
48 views

Proving that $f_n(x)=x^n$ does not converge on $[0,1]$ under the $L_1$ metric

We see that the pointwise limit is $f(x)=1$ if $x=1$, and $f(x)=0$, for $x\in[0,1)$. (our background space is $C[0,1]$) How would one go about proving if $f_n\to f$? My feeling is that we do not have ...
1
vote
0answers
25 views

Solve a problem of convergence of integral

We have F $\in$ $C(\mathbb{R}^N;\mathbb{R})$, $F\ge0$ and we have that $\int_{\mathbb{R}^N}Fdx<+\infty$. How can i prove the existence of a sequence $r_k\to+\infty$ such that $r_k\int_{\partial ...
2
votes
1answer
71 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 ...
2
votes
1answer
57 views

Do simple functions converge almost everywhere?

Assume there is a sequence of simple functions s.t.: $$\|\int(s_m - s_n)\mathrm{d}\mu\|\to 0$$ Does it follow that there is a subsequence which converges almost everywhere? (Note the order of modulus ...
2
votes
2answers
48 views

Boundedness of a sequence of functions

Let $\Omega$ be a bounded domain and $f_n\in L^2(\Omega)$ be a sequence such that $$\int_\Omega f_nq\operatorname{dx}\leq C<\infty\qquad \text{for all}\quad q\in H^1(\Omega),\ ...
0
votes
1answer
27 views

Convergence of Integrands and Integrals

Suppose $E \subset \mathbb{R}$ is compact. Is it possible to find a sequence of positive continuous functions $f_n: E \to \mathbb{R}$ such that for every $x \in E$ we have $$f_n(x) \to f(x)$$ for some ...
1
vote
1answer
25 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
2answers
71 views

Finding a limit of an integral

I am trying to find the following limit. Let $X = [0,\infty)$ and $\mathbb B$ denote the Borel subsets in $[0,\infty)$, $\lambda$ the Lebesgue measure. Let $f_n : [0, \infty) \to \mathbb R$ be given ...
0
votes
2answers
44 views

Better proof for: if $u_n \to u$ in $L^2$ then $F(u_n) \to F(u)$

Let $F(v) = \int_{A} v^2(x)J(x)dx$ where $J$ is bounded. If $u_n \to u$ in $L^2$ then I want to show that $F(u_n) \to F(u)$. The proof is $$F(u_n) - F(u) = \int_A (u_n^2 - u^2)J \leq C\int_A (u_n^2 ...
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
47 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 ...
5
votes
3answers
98 views

Lebesgue's Dominated Convergence Theorem problem

I am having trouble using DCT for the following Prove $$\lim_{n\to\infty}\int_0^\infty \frac{n}{(1+y)^n(ny)^\frac{1}{n}}dy = 1$$ I think most of the mass of the integral lies beneath $(e-1)/n$ but ...
0
votes
0answers
80 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 ...
4
votes
1answer
100 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 ...
0
votes
0answers
78 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 ...
2
votes
1answer
145 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 ...
4
votes
1answer
39 views

Limit outside compactly supported range converges to zero

Suppose $f\in C^\infty(\mathbb{R})$ is compactly supported on $[-N,N]$, and $1\leq p<\infty$, and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)\textrm dx=1$. Define ...
4
votes
1answer
32 views

Supremum over compactly supported range converges to zero

Suppose $f\in C^\infty(\mathbb{R})$ is compactly supported on $[-N,N]$, and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Define $$K_t(x)=\dfrac{1}{t}K\left(\dfrac{x}{t}\right)$$ It follows ...
3
votes
1answer
80 views

Bounding for convolution convergence

Suppose $f\in L^p(\mathbb{R})$ and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Define $$K_t(x)=\dfrac{1}{t}K\left(\dfrac{x}{t}\right)$$ I'm trying to prove that $\lim_{t\rightarrow ...
1
vote
0answers
123 views

If $\int(f_n) \rightarrow \int(f)$ then $\int(|f_n-f|) \rightarrow 0$ for $f_n \rightarrow f$ pointwise

I'd like to show that for an integrable sequence of functions $f_n:X \rightarrow [0, \infty)$ with $\sup_{n\geq 1} \int_{X} f_n d\mu < \infty, f_n \rightarrow f$ pointwise a.e. for some function ...
0
votes
1answer
117 views

Can I use this trick for my proof?

I am given the following problem, with a hint that states that I should use the General Lebesgue Dominated Convergence Theorem. Let $\{f_n\}$ be a sequence of integrable functions on $E$ for which ...
0
votes
1answer
44 views

What are some good integration problems where you can use some of the function convergence theorem of Lesbegue integrals?

I have learned about two major convergence theorem for the Lesbegue Measure: The monotone convergence theorem The dominated convergence theorem These are useful theorems for calculating integrals ...
2
votes
1answer
72 views

Confusion regarding proof using Fatou's lemma

This is in reference to the book Problems in Mathematical Analysis III by Kaczor and Nowak. We are given that ${f_n}$ converges to $f$ on $R$. Suppose that $\lim_{n \to \infty} \int_R{f_n dm} - \int_R ...
2
votes
1answer
54 views

Lebesgue's convergence for $H(u_n)\nabla u_n$ where $H$ is not everywhere defined

Consider the Heaviside function that is undefined in zero, i.e. $$H(t)=\begin{cases} 1&t>0 \\ 0&t< 0\end{cases}$$ Now consider a sequence of $H^1(\Omega)$-functions $u_n\to u$ in the ...
1
vote
1answer
114 views

Limit of sequence of integrals

$$\lim_{n\rightarrow \infty} \int _0^\infty \frac{(1-e^{-x})^n}{1+x^2}dx.$$ this is less then $$\lim_{n\rightarrow \infty} \int _0^\infty \frac{1}{1+x^2}dx.$$ so the limit should exist by DCT. But ...
5
votes
1answer
178 views

Showing a Lebesgue integral exists, while another doesn't.

Consider $$f_p(x)=x^p \exp\left(-x^8\sin^2x\right)$$ I have to show that $f_2\in\mathscr L(0,+\infty)$ whilst $f_3\notin \mathscr L(0,+\infty)$. Now, I am looking at the case $p=2$. The problematic ...
5
votes
2answers
155 views

For $f$ periodic, $g\to 0$ the integral of $fg$ converges (under some more conditions)

Let $f,g:\mathbb{R}\to\mathbb{R}$ be continuous functions such that: $f$ is periodic, with finitely many zeros in a period The average value of $f$ on a period is $0$ $g$ is monotonic decreasing and ...
1
vote
2answers
69 views

Convergence of $\int_{A_n} f$ to $0$

I am looking for a name or a reference in a textbook for the following result in order to quote it. For any $f\in L^1(\mathbb{R})$-integrable function, we have $$\lim_{n\to\infty}\int_{A_n} ...
0
votes
0answers
78 views

Integration by part for Lebesgue integral

I tried to prove one theorem in convergence of random variables and found myself in a little bit of trouble when doing integration by part. The reason being it involves a Lebesgue integral which I am ...
0
votes
1answer
146 views

Prove convergence of improper integral using change of variable.

This may be trivial, but I could use some help... Consider a real function $f: (0,1) \rightarrow \mathbb{R}$, continuous, positive, but not necessarily bounded. Let $g: [0,1] \rightarrow [0,1]$ be a ...
2
votes
1answer
88 views

Convergence and Lebesgue Integration

I came across this question in a textbook on introductory Lebesgue Integration. I have been teaching myself this material but was unsure of how to do the following question: Let $(g_n)$ be a sequence ...
4
votes
2answers
150 views

Question from Folland on modes of convergence

I have been reading through Folland, and I am having a hard time answering the following question. Any help will be much appreciated. Suppose $\lvert f_n \rvert \leq g \in L^1$ and $f_n \rightarrow ...
1
vote
1answer
176 views

About a proof that elements in a certain $L_2$ convergent series are also in $L_\infty$

The problem I have is about convergence of series expansions of random fields (or stochastic processes, if you will), which don't converge in the norm I want, that is $L_\infty$, but in $L_2$. I have ...
2
votes
1answer
136 views

Uniform convergence in $L^p$-spaces

Let $f\in L^p(0,\infty)$, $p>1$. Show that $\int_0^\infty f(x)\frac{\sin xy}{x} dx$ converges uniformly in $y$ in every finite interval. Show also that $|g(t+y)-g(y)|\leqslant M|t|^{\frac{1}{p}}$. ...
0
votes
1answer
106 views

About the absolute convergence of a series of measurable functions

Let $(X,\mathfrak M,\mu)$ be a measure space and $f_n:X\rightarrow [-\infty,+\infty]$ a measurable function for all $n\in\mathbb N$. Suppose that $$\sum_{n=1}^\infty ...
1
vote
1answer
266 views

Construction of a sequence of simple functions converging pointwise to a given function

Q1: How to construct a sequence of $\{f_n\}$ of simple function for a function $f$ such that $f_n\to f$ converges pointwise? Q2: If $f$ is measurable is $f_n$ also measurable for each $n$? Q2 is by ...
2
votes
2answers
119 views

$f_n \rightarrow 0$ a.e. on $[0,1]$ & $\int_{[0,1]} |f_n|^2 dm \leq 1$ $\implies$ $\int_{[0,1]} |f_n| dm \rightarrow 0$

Let $f_n : [0,1] \rightarrow \mathbf{R}$ be a sequence of measurable functions such that $\bullet$ $f_n \rightarrow 0$ a.e. on $[0,1]$. $\bullet$ $\int_{[0,1]} |f_n|^2 dm \leq 1$ for all $n \geq ...
10
votes
1answer
138 views

$\int_X |f_n - f| \,dm \leq \frac{1}{n^2}$ for all $n \geq 1$ $\implies$ $f_n \rightarrow f$ a.e.

Let $(X, M, m)$ be an arbitrary measure space. Let $f_n, f \in L^1_m(X)$. Assume that $$\int_X |f_n - f| \, dm \leq \frac{1}{n^2} \text{ for all }n \geq 1. $$ Then I want to show that $f_n ...
2
votes
1answer
134 views

Monotone convergence

Consider $X$ as non-decreasing non-negative function. Consider $\mu$ and $\nu$ as two probability measures on $(\mathbb{R},\mathcal{B})$ for which we know $\mu([t, \infty)) \geqslant \nu([t, \infty)) ...
8
votes
0answers
509 views

Difference of differentiation under integral sign between Lebesgue and Riemann

Here is a consequence of Lebesgue dominated convergence theorem on differentiation under integral sign. Function $f(x, t)$ is differentiable at $x_0$ for almost all $t \in A$, and $t \to f(x, t)$ ...
2
votes
1answer
546 views

The extension of Lebesgue Dominated Convergence Theorem

I want to prove that LDCT(Lebesgue Dominated Convergence Theorem) continues to hold if I replace the hypothesis $f_n \to f$ (convergence pointwise) with $f_n\to f$ (convergence in measure): $$\int ...
2
votes
1answer
533 views

Lebesgue Convergence using The General Lebesgue Dominated Convergence Theorem

Let ${f_n}$ be a sequence of integrable functions on E for which $f_n \to f$ a.e. on E and f is integrable over E. Show that $\int_E |f-f_n| \to 0$ if and only if $\lim_{ n\to\infty} \int_E |f_n| = ...
3
votes
2answers
340 views

The General Lebesgue Integral

For a measurable function, $f$, on $[1, \infty)$ which is bounded on bounded sets, define $a_n = \int_n^{n+1} f$ for each natural number $n$. Is it true that $f$ is integrable over $[1, \infty)$ if ...
3
votes
1answer
4k views

{$\int_{[1/n,1]}f$} to converge and yet $f$ is not $L$-integrable over $[0,1]$

Let $f$ be a function on $[0,1]$ and continuous on $(0,1]$. I want to find a function $f$ s.t. {$\int_{[1/n,1]}f$} converges and yet $f$ is not $L$-integrable over $[0,1]$. My attempts: I've found ...