All Questions
64
questions
4
votes
1
answer
96
views
Question about Proof of the Integrability of $f$ and $f_1,f_2,\dots,$ in Lebesgue's Dominated Convergence Theorem
I am self-studying measure theory and got stuck on part of the proof of the Lebesgue's Dominated Convergence Theorem:
Theorem$\quad$ 2.4.5$\quad$ (Lebesgue's Dominated Convergence Theorem) Let $(X,\...
- 2,493
3
votes
0
answers
50
views
Proof of Beppo Levi's Theorem [closed]
I am self-studying measure theory using Measure Theory by Donald Cohn. The text presented the following result but lack of detailed proof. I tried to write up the proof, and I would really appreciate ...
- 2,493
2
votes
1
answer
41
views
A question about switching the order of $\int$ and $\lim$ for a series of complex functions
When I took undergraduate complex analysis, the instructor was trying to prove an inequality, and he used a technique as below:
$$\int \lim_{n\rightarrow \infty} f_n dz= \lim_{n\rightarrow \infty} \...
- 1,131
0
votes
1
answer
35
views
$\lim_{x \to \infty} \frac{\int^x_1 t^{-1} \rho(t) dt }{\int^x_1 t^{-1} dt} = \rho$, where $\lim_{t \to \infty} \rho(t) = \rho \in (0,\infty)$
I want to prove that $\lim_{x \to \infty} \frac{\int^x_1 t^{-1} \rho(t) dt }{\int^x_1 t^{-1} dt} = \rho$, where $\lim_{t \to \infty} \rho(t) = \rho \in (0,\infty)$ with $\rho \colon (0,\infty) \to (0,\...
- 1,710
0
votes
0
answers
56
views
Why if $\phi \in L^1(\mathbb{R}^d)$ then $\int_{|x|>R} |\phi| \ dx < \epsilon $
Why if $\phi \in L^1(\mathbb{R}^d)$ then $\displaystyle\int_{\|\mathbf{x}\|>R} |\phi|\mathrm{d}\mathbf{x} < \epsilon $?
I started the proof by:
Suppose that $\exists \epsilon_0 > 0$ and $j_0 \...
- 1
4
votes
1
answer
183
views
Almost Dominated Convergence Exercise (not Generalized Dominated Convergence Theorem)
I just had a question regarding an exercise from Tao's Intro to Measure Theory (Exercise 1.4.46). The question is as follows:
Let $(X, B, \mu)$ be a measure space, and let $f_1, f_2, ...: X \to C$ be ...
- 43
5
votes
1
answer
172
views
Finding $\lim_{n\to\infty} \int_0^{\infty}\frac{1}{(1+\frac{x}{n})^nx^{\frac{1}{n}}}\, d\lambda$
The question
Compute the limit:
$$\lim_{n\to\infty} \int_0^{\infty}\frac{1}{\displaystyle\left(1+\frac{x}{n}\right)^nx^{\frac{1}{n}}}\ \mathrm{d}\lambda$$
My attempt
It is clear to see that point-wise ...
- 1,847
1
vote
2
answers
96
views
How to prove that $\int_{[-\pi,\pi]}\log(\vert 1- \exp(it)\vert)\mathrm{d}\lambda(t)=0$?
Let $r>1$ and $\displaystyle\int_{[-\pi,\pi]}\log(\vert 1- r\exp(it)\vert)\mathrm{d}\lambda(t)= 2\pi\log(r)$.
We want to prove that :
$\displaystyle \lim \limits_{r\to 1} \int_{[-\pi,\pi]}\log(\...
- 3,330
0
votes
1
answer
47
views
Does $\lim_{a\to 0}\int_1^\infty f\left(\sqrt{a^2+x^2}\right)dx=\int_1^\infty f\left(x\right)dx$ for smooth, rapidly decaying $f$?
Let $f:[1,\infty)\to\mathbb R$ be infinitely smooth and suppose $f$ and its derivatives decay faster than any power of $x$ as $x\to\infty$. In other words, $f\in\mathscr S([1,\infty)),$ where $\...
- 6,672
0
votes
2
answers
65
views
$\lim_{y \to 0}\frac{1}{\ln(y)}\int_{y}^1\frac{1-\cos(x)}{x^3}dx=-1/2$
We want to find an equivalent to $\int_{y}^1\frac{1-\cos(x)}{x^3}dx$ when $y \to0.$ It seems that $\lim_{y \to 0}\frac{1}{\ln(y)}\int_{y}^1\frac{1-\cos(x)}{x^3}dx=-1/2.$
After Trying an integration by ...
- 1,070
2
votes
3
answers
123
views
Find $\lim_{n \rightarrow \infty} \int_0 ^{\infty} \left(1+ \frac{x}{n}\right)^n e^{-ex} dx$ [duplicate]
Find $$\lim_{n \rightarrow \infty} \int_0 ^{\infty} \left(1+ \frac{x}{n}\right)^n e^{-ex} dx.$$
I know that $\lim_{n \rightarrow \infty} \left(1+ \frac{x}{n}\right)^n = e^x.$
Also, I showed that $ \...
- 195
0
votes
3
answers
124
views
Evaluate $\lim_{n \rightarrow \infty} \int_0^1 \frac{nx^{n-1}}{2+x} dx$
Evaluate $\lim_{n \rightarrow \infty} \int_0^1 \frac{nx^{n-1}}{2+x} dx$.
Attempt:
Let $h(x)=x^n$. Then $nx^{n-1} = h'(x)$.
I'm thinking about using the Dominated Convergence Theorem.
So $f_n= \frac{nx^...
- 663
0
votes
1
answer
84
views
$E \xi_n^2 \le c $ and $\xi_n \to \xi$ almost surely. Prove that $E \xi $ is finite and $E \xi_n \to E \xi.$
Suppose $\{\xi_n\}$ is a sequence of random variables on a probability space such that $E \xi_n^2 \le c $ for some constant $c.$ Assume that $\xi_n \to \xi $ almost surely as $n \to \infty$. Prove ...
- 5,230
3
votes
1
answer
142
views
Finding $\lim_{n \to \infty} \int_{2}^{\infty} \frac{n\sin\left(\frac{x-2}{n}\right)}{(x-2)+(1+(x-2)^2)} dx$
I have to calculate the following limits, using a theorem but I don't really know what theorem to use (it is for the subject of measurement and integration, for the unit "Measurable functions, ...
- 953
0
votes
1
answer
102
views
Limit of integral of measurable function
Let $(X, \mathcal{A}, \mu)$ be a measurable space and f a non negative measurable function. Let $E=\{x \in E: f(x) < 1\}$. Calculate $$\lim_{n \to \infty} \int_E e^{-f^n} d\mu.$$
I'm having some ...
