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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2
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0answers
36 views

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

$B=\{(x,y,z) \in\mathbb R^3| \; x^2+y^2+4z^2 \le3, \;x^2-y^2+4z^2\le1, \; z\ge 0\}$ The question is similar to that which I shared in another topic. Also here, the set is defined by an ellipsoid, ...
4
votes
2answers
94 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 ...
1
vote
0answers
38 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 $M$...
2
votes
3answers
64 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 ...
1
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1answer
45 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 $\boldsymbol{r}\in\mathbb{R}^3\setminus\overline{V}$ be any point external to $V$. I intuitively suppose that ...
4
votes
0answers
28 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 ...
0
votes
0answers
34 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 ...
1
vote
0answers
30 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^...
3
votes
2answers
52 views

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 ...
2
votes
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 $ ?
2
votes
1answer
37 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)...
1
vote
0answers
32 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}^{...
2
votes
2answers
37 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 ...
1
vote
1answer
45 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)....
1
vote
1answer
46 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 $\int f_n\,d\lambda =1$, ...
1
vote
2answers
59 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 ...
4
votes
2answers
66 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^\...
0
votes
0answers
51 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 $A$ ...
1
vote
0answers
43 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 > -\...
1
vote
2answers
47 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} \...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
2
votes
1answer
61 views

How to apply Fubini's theorem?

I was asked to show the equality of these integrals $$\int_0^1\int_0^1\frac{x^2-y^2}{(x^2+y^2)^{3/2}}\log(4+\sin x)dydx =\int_0^1\int_0^1\frac{x^2-y^2}{(x^2+y^2)^{3/2}}\log(4+\sin x)dxdy\tag{1}$$ ...
0
votes
0answers
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 ...
5
votes
0answers
63 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}{\...
0
votes
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 ...
0
votes
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 $E(f(x))$. ...
1
vote
1answer
54 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 $[...
1
vote
1answer
31 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 \...
1
vote
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+...
1
vote
2answers
87 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 ...
0
votes
0answers
32 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)...
1
vote
0answers
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 ...
1
vote
0answers
13 views

Can distribution function of $f$ be expressed as triangle inequality form?

Let $w_f$ be a distribution function of $f$ on $E\in\Bbb{R^n}$,$$w_f(\alpha) := \mu\left(\{\mathbf{x}\in E ~|~ f(\mathbf{x} \gt \alpha\}\right)$$ From triangle inequality, $$|f| \le |f-f_k|+|f_k|.$$...
1
vote
1answer
22 views

Show $f$ is integrable over $[1, \infty)$ iff $\sum_{n=1}^{\infty}\left|\int_n^{n+1}f \right|$ converges

For a measurable funtion on $[1,\infty)$ which is bounded on bounded sets define $a_n=\int_n^{n+1}f $. Is it true that $f$ is integrable over $[1,\infty)$ iff the series $\sum_na_n$ converges ...
3
votes
0answers
25 views

Continuous function but iterated integrals are not equal

Given a sequence $(g_n)$ of continuous functions $g_n:\mathbb R \to [0,\infty)$ with the properties $\operatorname{supp} g_n^{(1)} \subset (n,n+1)$ and $\int g_n \, d\lambda=1$ for all $n$, how can I ...
1
vote
1answer
39 views

Dominated convergence theorem exercise application

I have to prove that $$\lim_{n \to \infty} \int_0^\infty \mathrm (1+x/n)^{-n}(x^{-1/n})\mathrm{d}x = 1$$ I've been told to use Dominated convergence theorem but I can't find a function $|f_n(x)| \le ...
1
vote
1answer
68 views

Not integrable although iterated integrals are equal

How can I show that the function $$f=\begin{cases} 0 & (x,y)=(0,0)\\\frac{xy}{(x^2+y^2)^2} & \mbox{else}\end{cases}$$ is not Lebesgue-integrable, although the iterated integrals exist and are ...
1
vote
1answer
26 views

Using Tonelli - upper and lower limits of the integral

Given $$f(x,y)=x^2+y^2$$ and a triangle $A$ with the corner points $$(0,0),(0,1),(\frac{1}{2},\frac{1}{2})$$ how do I compute $$\int_A fd\lambda^2$$ I know I have to use Tonelli, but do I have to ...
2
votes
1answer
43 views

An uncountable chain of equivalence relations

First, an example: We know that, for two real valued, Lebesgue-integrable functions, the relation "equals almost everywhere" is an equivalence relation. In particular, if $f_0$ is Lebesgue-integrable, ...
4
votes
1answer
49 views

A Measure Theory problem-If $\int_{A_n}f(x)dx\rightarrow0 $ then $\lambda(A_n)\rightarrow0$

This question was proposed as part of a test for PhD applicants but considered too hard and rejected. I tried unsucessfully to solve it for quite some time. For anyone wishing to try his luck.. ...
2
votes
1answer
49 views

Is $\iint \dfrac{1}{z} dxdy\neq 0$?

I am trying to solve an exercise and at some point I came accross the integral $$\iint_L \dfrac{1}{z} dxdy,$$ ($z=x+iy$) where $L\subset \mathbb{C}$ is a compact set with positive two-dimensional ...
1
vote
1answer
43 views

Is it true that $\lim\limits_{r \rightarrow 1}f(r x) = f(x)$ in $L^1$?

Suppose $f \in L^1(\mathbb{R})$ with Lebesgue measure and $r > 0$. Does $f(rx)$ converges to $f$ in $L^1$ as $r \rightarrow 1$ ? Put differently, does $$ \| f(rx) - f(x)\|_1 \rightarrow 0$$ as $r \...
0
votes
1answer
26 views

Doubts concerning to an application of Frullani's theorem to $f_k(x)=\frac{2^{-k^2x}-2^{-(k^2+1)x}}{x}$, and Lebesgue convergence theorems

By application of Frullani's theorem for $a_n=n^2+1$, $b_n=n^2$ where $n\geq 2$ and $f(x)=2^{-x}$ then RHS in Frullani's integral is obtained for $n\geq 2$ as $$\log(1-\frac{1}{n^2}),$$ thus I asked ...
0
votes
1answer
44 views

Bounding Problem /Conditions for the Lebsgue Integral of a Function depending on two parameters to be continuous

let $F(t)=\int_{E} f_t(x)$ for $t\in J \subseteq \mathbb{R}$. Then some theorem says that $F$ is continuous if $1)$ $\forall t_0$ $f_t(x) \rightarrow f_{t_0}(x)$ as $t \rightarrow t_0$ almost ...
5
votes
1answer
72 views

Prove that $f\in L^2$ and $\lim_{n\rightarrow\infty} \int_A f_n = \int_Af$

Let $A$ be a bounded, measurable susbset of $\mathbb{R}$. Prove that if $(f_n) \subset L^2 (A)$ converges uniformly to $f$ on $A$, then $f\in L^2(A)$ and $\lim_{n\rightarrow\infty} \int_A f_n = \...
1
vote
2answers
24 views

For what $q$ is $\frac{\sin(x)}{x^q}$ Lebesgue Integrable on (0,1] where $q>0$

You can show $\frac{1}{x^q}$ converges on$ (0,1] $ for $q<1$ and that's a bound for the $\frac{|\sin(x)|}{x^q}$ so we know for $q<1$ our function is integrable- I can't seem to improve on this ...
2
votes
0answers
29 views

Is an integrable function always measurable?

There is a theorem in my textbook. Theorem $5.1$ Let $f$ be a nonnegative function defined on a measurable set $E$. Then $\int_E f$ exists if and only if $f$ is measurable. Notation If $\int_E f$...
0
votes
1answer
70 views

what does $\frac{\text{d}x}{x}$ mean?

I saw in a lecture recently the Gamma-function written like $$\Gamma (k) = \int_0^\infty e^{-x} x^k \frac{\text{d}x}{x}$$ and the professor said, that the integral was with respect to the measure $\...