For question about integration, where the theory is based on measures. So it's almost always used together with the tag [measure-theory], and its aim is to specify questions about integral, not only properties of the measure.

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How to start an eigenvalue problem

I am stuck on this problem : This is an eigenvalue problem $$\phi''+ \lambda^2 x(x+2)^2 \phi =0\\\phi(1)=0\\ \phi(0)=0$$ I forget this kind of problems... please give me a hint or a clue ,cause I ...
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38 views

how to prove $f_n \in L^1$

I was trying to build a scheme to solve this kind of question: Let D be a domain of $\Bbb R^n$ anf $f_n$ : D $\to$ $\Bbb R$. Say if $f_n \in L^1(D)$ First of all I need to check that both $f_n$ anf $...
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1answer
47 views

Show that integral is analytic

Let $h:[0,\infty)$ be an integrable function. Prove that the function $$g(z)=\int_0^\infty h(t)e^{tz}\,dt$$ is analytic on $\{z=x+yi:x<0,y\in\mathbb{R}\}$. How do I start for this question? I ...
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19 views

Integral of magnetic field inside cylinder

Let $V\subset\mathbb{R}^3$ be an infinitely high solid cylinder, or a cylindrical shell of radii $R_1<R_2$, whose axis has the direction of the unit vector $\mathbf{k}$. For any point of ...
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1answer
22 views

Dilation convergence in L^1

Below is a question, which I asked before, from Stein's Real Analysis. I've provided a partial solution, which I think it's pretty along the lines of what needs to be done, however, I have no ...
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1answer
407 views

limits of an integrable function over increasing sequence of sets

Let $(E_n)_{n \geq 1}$ be an increasing sequence of sets such that $\bigcup_{n \geq 1} E_n = \Omega$. Then for every integrable function $f$ we have $$\lim_{n \rightarrow \infty} \int_{E_n} f d\mu = \...
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2answers
89 views

How can I solve this triple integral $\iiint_{B} y\;dxdydz$ on a defined set?

Calculate $$\iiint_{B} y\;dxdydz.$$ The set is $\;B=\{(x,y,z) \in \mathbb R^3$; $\; x^2+y^2+4z^2\le12$, $-x^2+y^2+4z^2\le6$, $y\ge 0 \}$. I know that B is defined by a real ellipsoid, an ...
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46 views

If $\{f_n\}\subset L_1([0,1])$, $f_n\to f$ pointwise, and $\sup_{n} \int_{0}^{1} |f_n|\max (0, \log |f_n|)<\infty$, then $f_n\to f$ in $L_1$

I'm going through old analysis qualifying exams, and have come to a roadblock on the following problem: Suppose that $\{f_n\}\subset L_1([0,1])$, $f_n\to f$ pointwise, and $\sup_{n} \int_{0}^{1} |f_n|...
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Justify $\lim_{n\to\infty}n^p\int_0^1\sum_{k=n}^\infty\frac{\sigma(k)e^{x/k}}{k^{p+2}\log\log k} dx=\frac{e^\gamma\int_0^1f(x)dx}{p}$

Inspired in PROBLEM 207, La Gaceta de la Real Sociedad Matemática Española, Vol. 16, N0. 3 (page 507 in spanish, proposed and solved by Furdui), I've tried write examples of this new statement ...
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28 views

Lebesgue integral question (double integral)

Let $g,h$ be nonnegative Lebesgue measurable functions on $\mathbb{R}$. Prove that $$\int_{-\infty}^\infty g(x)^2h(x)\,dx=\int_0^\infty\int_{\{t\in\mathbb{R}:g(t)>x\}}2h(t)x\,dtdx.$$ I am lost on ...
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34 views

How can I find the measure of $B=\{(x,y,z) \in\mathbb R^3| \; x^2+y^2+4z^2 \le3, \;x^2-y^2+4z^2\le1, \; z\ge 0\}$? [on hold]

$B=\{(x,y,z) \in\mathbb R^3| \; x^2+y^2+4z^2 \le3, \;x^2-y^2+4z^2\le1, \; z\ge 0\}$ The question is similar to that which I shared in another topic. Also here, the set is defined by an ellipsoid, ...
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55 views

Collection is uniformly integrable, but individual is not integrable

Could you give me an example about: "a collection of functions that is uniformly integrable but each (or some) function in the collection is not integrable." This sounds counterintuitive? However ...
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32 views

Bounding $L_p$ norms on a convergent $L_1$ sequence

I've encountered a prelim problem on $L_p$ spaces that I'm pretty stuck on. Suppose $1 < p < \infty$ and $f_n \in L_1([0,1]) \cap L_p([0,1])$, with $||f_n||_p$ bounded above by some constant $M$...
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59 views

Is it correct to interpret the “dx” in the standard notation for integrals as the Lebesgue measure?

Ok so I am in my Calc I class for the summer and we are just beginning to talk about integrals. I know a little bit about measure theory and the Lebesgue integral and why is it more general than the ...
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1answer
42 views

Lebesgue integral of $\frac{1}{\|\boldsymbol{x}-\boldsymbol{r}\|^2}$ on an infinite cylinder

Let $V\subset \mathbb{R}^3$ be a solid infinite cylinder, or cylindrical shell, and let $\boldsymbol{r}\in\mathbb{R}^3$ be any point of the space. I intuitively suppose that the Lebesgue integral $$\...
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26 views

On sharp bounds of some dyadic operators

I am now having interest in finding the sharp bounds of some kinds of dyadic operators which map $L^p(\mathbb{R})$ to $L^p(\mathbb{R})$. For example, the martingale transform $T_{\sigma}$ which ...
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2answers
56 views

Proofs related to chi-squared distribution for k degrees of freedom

I was reading a proof related to chi-squared distribution for k degrees of freedom from wiki. https://en.wikipedia.org/wiki/Proofs_related_to_chi-squared_distribution I think I might understand the ...
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32 views

Understanding a calculation deduced for the function $\pi^{-s/2}\Gamma(s/2)\zeta(s)$

With my current knowledges I don't know if this is a bad question, but since I am interesting in this kind of calculations I want to ask you, if I was wrong or if if my statement is obvious. From ...
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29 views

If $\|g_1 - g_2\|_{\infty} = 0$ then $\int g_1 f\ d\mu = \int g_2 f\ d\mu$ for all $f \in L^1$

I am reading Cohn, Measure Theory, 2nd edition. Proposition 3.5.5 states that if $(X, \mathcal{A}, \mu)$ is a measure space and $1 \leq p < \infty$ and $1/p + 1/q = 1$, then the map $T : L^q \to (L^...
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Frullani's theorem in complex context, other examples

One has as application of Frullani's theorem in complex context that $$\int_0^\infty \frac{e^{-x\log 2}-e^{-xb}}{x}dx=\mathcal{Log} \left( \frac{1}{2\log 2}+i\frac{B}{\log 2} \right) $$ where I taken ...
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2answers
77 views

$L^p \subset L^q$

Let $(X,M,\mu)$ be a measure space. Let $\Omega \subset X$ be a measurable set. We have $L^2(\Omega) \subset L^1(\Omega)$ . Can we have that $\mu(\Omega)< \infty $ ?
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1answer
410 views

Holder inequality (reverse or equality?)

For bounded $\Omega\in\mathbb{R}^n$, it is easy to see by the Holder inequality that $\int_{\Omega} u\,dx\leq (\int_{\Omega} 1^2\,dx)^{\frac{1}{2}} (\int_{\Omega} u^2\,dx)...
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1answer
35 views

Differentiation under the integral sign, where the partial derivative of the integrand is not bounded by a Lebesgue integrable function.

Let $K(t)=\int_1^\infty u(t,x)\ \mathrm{d}x$, where $$u(t,x)=\frac{\cos{tx}}{x^2}\mathbb{1}_{[1,\infty)}(x).$$ I need to show that, for $t>0$, $$\frac{dK}{dt}(t)=\frac{1}{t}\left(K(t)-\cos{t}\right)...
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30 views

Well-definedness of Fourier transform of $f\in L^1(\mathbb{R})$

Let $f\in L^1(\mathbb{R})$ where the measure is taken to be the Lebesgue measure. The Fourier transform of $f$ is the function $\hat{f}$ defined as $$\hat{f}(\xi)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{...
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36 views

Bound on integrable nonnegative function $F$ given inequality with compactly supported continuous functions.

Full Question: Suppose that $F$ is a nonnegative function that is integrable on $\mathbb R$ and there is a constant $C$ such that $\int_\mathbb R Ff \leq C\int_\mathbb R f$ whenever $f$ is a ...
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1answer
43 views

Different approaches to differentiability in $L^2$

We can use different approaches to differentiability of $L^2(\mathbb{R})$ functions, e.g. we can say that $f\in L^2(\mathbb{R})$ is differentiable iff $f$ has a differentiable version (representative)....
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1answer
45 views

Where $\{q_n\}=\mathbb Q$ and $f_n:[q_n-2^{-n-1},q_n+2^{-n-1}]\to[0,\infty)$ with $\int f_n\,d\lambda=1$, show $\sum_{n=1}^\infty f_n<\infty$ a.e.

That is: Let $\mathbb Q=\{q_n\}_{n\in\mathbb N}$ be an enumeration of the rationals. Let $f_n$ be a nonnegative Borel measurable function supported on $q_n\pm 2^{-n-1}$ with $\int f_n\,d\lambda =1$, ...
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61 views

Does existence of the second weak derivative of $f\in L^2$ imply existence of the first?

Let's consider a function $f\in L^2(\mathbb{R})$ for which the second weak derivative exists and lie in $L^2(\mathbb{R})$, i.e. there exists $f''\in L^2(\mathbb{R})$ such that for all $\varphi\in C_0^\...
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47 views

When we can permute between the integral and convex hull?

Is there a relation between the following expressions? $$\operatorname{conv}\left(\int_{0}^{t} f(s,x)ds :x \in A \right) $$ and $$\int_{0}^{t} \operatorname{conv}(f(s,x):x \in A)\ ds $$ where $A$ ...
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1answer
645 views

Why is Lebesgue-Stieltjes a generalization of Riemann-Stieltjes? Moreover, is there an example where Lebesgue-Stieltjes is useful

I certainly have a question, but i don't know what the best title should be. Please edit the title if there is a better one :) And I believe, to get a better answer, it would be good to explain ...
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1answer
61 views

Evaluation of $\int_{[0,\infty)}\biggl(\int_{[0,\infty)}2x\sqrt{y}e^{-x^2\sqrt{y}-y}dy\biggl)dx$.

Find the value of $$\int_{[0,\infty)}\biggl(\int_{[0,\infty)}2x\sqrt{y}e^{-x^2\sqrt{y}-y}dy\biggl)dx$$ My attept: Let $f(x,y)=2x\sqrt{y}e^{-x^2\sqrt{y}-y}$ to apply Tonelli theorem I did the ...
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199 views

Prove that the Lebesgue integral of $f\chi$ equals to $0$ indicates that $f=0$ a.e.

Suppose $f: [0,1] \to \mathbb{R}$ is bounded, measurable, and $$\int_{[0,1]}f \chi_{[0,a)}\, d\mu = 0$$ for all $a \in [0,1]$. Prove that $f=0$ a.e. I know that if $\int_{[0,1]}f\, d\mu = 0$,...
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0answers
42 views

Lebesgue integral of $-\frac{1}{n}$

I have following sequence: $$ f_n = -\frac{1}{n} $$ I wanted to show, that it converges to $f = 0$, but my book says, that it doesn't, because the condition $$ \int_{\mathbb R} f_1 d\lambda > -\...
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293 views

Prove series converge for almost every $x$

Let $f\in L^p(\mathbb{R})$, $1<p<\infty$, and let $\alpha>1-\frac{1}{p}$. Show that the series $$\sum_{n=1}^{\infty}\int_n^{n+n^{-\alpha}} |f(x+y)|dy$$ converges for a.e. $x\in \mathbb{R}$. ...
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46 views

Questions on measurable sets

I'm learning about measure theory, specifically measurable sets, and need help with the following exercises: $(1)$ Find the measure of the set $E_1 = \mathbb{Z} \cup \mathbb{Q} \cup (\mathbb{R} \...
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1answer
21 views

$L^1$ approximation by a slightly “displaced” copy

Let $f:\Bbb R\to \Bbb R$ be an $L^1$ function and $f_\epsilon(x):=f(x+\epsilon)$, $\mu$ is the Lebesgue measure, prove that $$\lim_{\epsilon\to 0}\int|f_\epsilon-f|\mathrm d\mu=0.$$ I tried to ...
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1answer
49 views

How to use Cavalieri?

How can I compute the volume of $$S=\{(x,y,z)\in\mathbb R^3\, |\, x^2+y^2 \le\frac{1}{(1+z)^2}, 0\le z\le 1\}$$ by exclusively using integration? I know that I can use Cavalieri, but I don't ...
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1answer
61 views

How to apply Fubini's theorem?

I was asked to show the equality of these integrals $$\int_0^1\int_0^1\frac{x^2-y^2}{(x^2+y^2)^{3/2}}\log(4+\sin x)dydx =\int_0^1\int_0^1\frac{x^2-y^2}{(x^2+y^2)^{3/2}}\log(4+\sin x)dxdy\tag{1}$$ ...
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54 views

$f \in L^1(\mathbb{R})$ implies there exists $a \in \mathbb{R}$ with $\int_{(-\infty, a]} f = \int_{[a, \infty)} f$

In studying for a qualifying exam, I found a problem asking me to prove: If $f \in L^1(\mathbb{R})$ has $\int_\mathbb{R} f \neq 0$, then there exists $a \in \mathbb{R}$ with $\int_{(-\infty, a]} f ...
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1answer
52 views

Is this family of functions uniformly integrable over $[0,1]$

Let $\mathcal F$ be a family of functions on $[0,1]$ each of which is integrable over $[0,1]$ and has $\int_a^b|f|\le b-a$ for all $[a,b] \subseteq [0,1]$. Is $\mathcal F$ uniformly integrable over $[...
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1answer
76 views

Understanding principal value integral

I'm reading the original article on distance covariance (link), and throughout the article the author uses the following lemma: Can someone please explain what he actually means by "principal value ...
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0answers
61 views

Difference between $d\mu(x)$ and $\mu(dx)$

In my lecture notes of probability course I found two different notations involving $d,\mu$ and $x$: is there any difference between $\mu(dx)$ and $d\mu(x)$? For example I read $\mu(dx) = \frac{1}{\...
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2answers
86 views

A $\sigma$-finite Borel measure $\mu$ on $\mathbb R$ with $\mu(\mathbb R)=\infty$ s.t. $\int f \,d\mu=\int g \,d\mu \implies f=g$ pointwise?

This is the final part of a problem on an old Analysis preliminary exam at my institution. We are given that $f,g\in L^1(\mathbb R,\mu)$ and $f\leq g$. For earlier parts of the problem, I've already ...
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1answer
56 views

Interchanging Expectation and Derivative

Suppose I have a random function, $f(x)(\omega)$. And that for fixed $\omega$, we have the derivative $g(x)(\omega)=\frac{d}{dx}f(x)(\omega)$. For a fixed $x$, I can find the expectation $E(f(x))$. ...
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1answer
52 views

Can Lebesgue Dominated Convergence always be used?

Suppose I want to find the derivative $$\frac{d}{dx}\int f(x,y) dy.$$ I want to know under what condition it would be equal to $$\int \frac{d}{dx}f(x,y) dy.$$ Of course, if I can find a suitable ...
3
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1answer
72 views

Let $f_n(x) = nx^{n-1}-(n+1)x^n$, $x\in (0, 1)$. Then $\int_{(0, 1)}\sum_{n=1}^{\infty}f_n \neq \sum_{n=1}^{\infty}\int_{(0, 1)}f_n.$

I'm learning about measure theory, specifically Lebesgue integration, and need help to understand the solution to the following problem: Let $f_n(x) = nx^{n-1}-(n+1)x^n$, $x\in (0, 1)$. Show that $$...
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1answer
30 views

Calculate the next limit. Lebesgue integral.

I'm trying to solve the next problem: Calculate, justifying all steps, the limit $$ \lim_{n \rightarrow \infty} \int_A \dfrac{1+ \dfrac{\cos^2(x^3)}{n} }{x^2+y^2+1} dx \ dy$$ where $A=\{(x,y) \in \...
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0answers
14 views

Suppose that there is a finite $c$ such that $\int_0^1 | f(a+t) - f(b+t) | dt \le c$ for all $a$ and $b$. Show that $f \in L(0, 1)$.

Please help me understand the following proof. Q) Let $f$ be measurable and periodic with period $1$, that is, $f(t+1)=f(t)$. Suppose that there is a finite $c$ such that $$\int_0^1 | f(a+t) - f(b+...
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0answers
31 views

Bounding $L^p$ norm of a function defined by averaging

Let $\Delta=\{t_0, t_1, ... t_m\}$ be a partition of $[a, b]$ and let $f{\in}L^{p}[a, b]$ for $1\le p\le\infty$. Let $T\Delta$ be the function on $[a, b]$ defined by $T\Delta(f)(a)=0$ and $$T\Delta(f)...
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18 views

Fatou lemma and weak convergence in Hilbert

In a Hilbert space $H$ a sequence $(x_n)_{n\geq0}$ is said to converge weakly to $x$ if $\forall y\in H:\langle y,x_n\rangle\rightarrow\langle y,x\rangle$, the case in which we can easily deduce an ...