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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1answer
46 views

Calderón-Zygmund theorem doesn't seem has correct hypotesis

This theorem states that, given ANY $f\in L^1(\Bbb R^n)$ and ANY $\alpha>0$, there exists a sequence of (mutual disjoint open with sides parallel to the axis) cubes $\{Q_k\}_{k\ge1}$ such that $$ ...
0
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1answer
27 views

What's the adjoint for the evaluation operator?

What's the adjoint for the evaluation operator, $A\in L([X\rightarrow Y],Y)$, where $Af=f(x)$ for some fixed $x\in X$? In case, there's any ambiguity, I'm looking for an operator $A^*$ such that ...
0
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1answer
17 views

Let $a_n$ be a sequence of nonnegative real numbers. Let $E = [1,\infty)$, and $f =a_n$ if $n\leq x<n+1$.Show that $\int_Ef = \sum a_n$

I have the following problem; Let $\{a_n\}$ be a sequence of nonnegative real numbers. Define the function $f$ on $E = [1,\infty)$ by setting $f(x) = a_n$ if $n\leq x<n+1$. Show that $\int_Ef = ...
0
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2answers
56 views

If $f$ is a bounded function on an interval E, and E has measure $0$ , Is $f$ measurable? What is the value of it's $\int_E f$?

If $f$ is a bounded function on an interval E, and E has measure $0$, Is $f$ measurable? What is the value of $\int_E f$? I have the question above in Royden Analysis 4e. Intuition suggests that f ...
0
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1answer
41 views

Is $\int f$ on [0,1] always equal to $\int f$ on [x,1] when take the limit of x to 0

Is $\int f$ on [0,1] always equal to $\int f$ on [x,1] when take the limit of x to 0? I know that if f is nonnegative, then I can use LMCT to prove it. However, how about f is only bounded, or only ...
2
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1answer
40 views

Why does $f(0)=\int f(x) \, d\mu =\frac14$ and $f$ extremal at $0$ imply $\mu(\{0\})=1$?

I just read a paper by Abner Shimony (The Status of the Principle of Maximum Entropy), and he is making a claim about a Lebesgue integral in appendix A that I don't fully understand yet. Let $f(x)$ ...
2
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1answer
88 views

Show that the integral of a bounded measurable function of finite support is properly defined.

Edit: My post might be lacking context. The question was posed because the book has only defined the Lebesgue integral of functions on a measurable set $E$ where $m(E)<\infty.$ The definition below ...
0
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0answers
30 views

A decreasing sequence of integrable functions with the limit of the integral exist

Question: Prove or disprove: If {${f_{n}}$} is a decreasing sequence of integrable functions such that $lim\int f_{n}$ exists in $\Bbb R$, then $lim\int f_{n}=\int lim\ f_{n}$. Attempt: I think this ...
0
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3answers
57 views

What is the integral of the function $f = \chi_{[0,\infty)}e^{-x}$

I have $$\int_{\mathbb R} f\,d\mu = \lim_{n\to\infty}\int_{[0,n]} f\,d\mu$$ So $$\begin{align}\int_{\mathbb R}f\,d\mu &= \lim_{n\to\infty}\int_{[0,n]}\chi_{[0,\infty)}e^{-x}\\ ...
2
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1answer
25 views

Why does this function preserve measure of null sets?

In this question, one of the answers claimed that the function $f: \mathbb{R}^{2n} \to \mathbb{R}$ given by $f(x,y) = x-y$ pulls back Lebesgue null sets to null sets, that is, $f^{-1}(N)$ is a null ...
3
votes
1answer
44 views

Monotone Convergence Theorem and Completeness of measurable space

I am reading about the monotone convergence theorem which states: Let $(X,\Sigma,\mu)$ a complete measurable space. If $f_n\rightarrow f$ monotone increasing and for almost all $x\in D$ then ...
0
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1answer
26 views

Bounded cube dense in $R^3$

I have a sequences of function $(u_n)_{n\in \mathbb{Z}}: \mathbb{R}^3\times [0,\infty)\rightarrow \mathbb{R}^3$ that satisfies \begin{equation}\lim_{n\to \infty}\int_{B(0,R)} ...
1
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1answer
116 views

Proof of fundamental theorem of integral calculus

This is the result I am after. Theorem Let $F:[a,b]\to\mathbb{R}$. Then the following are equivalent: $F$ is absolutely continuous, that is, for all $\epsilon$ there exists $\delta$ ...
0
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1answer
30 views

Extending the legitimacy of a type of differentiation under the integral to a larger class of functions

I know that If $f\colon \mathbb{R}^n \to \mathbb{R}$ is integrable on any measurable, according to the usual $n$-dimensional Lebesgue measure $\mu_y$, and bounded subset of $\mathbb{R}^n$, and if $g ...
0
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3answers
41 views

Increasing sequence of step functions and diverging Lebesgue Integrals

Let $\{s_n\}$ be an increasing sequence of step functions which converges pointwise on an interval $I$ to a limit function $f$. If $I$ is unbounded and if $f(x) \geq 1$ everywhere on $I$ except on a ...
0
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1answer
41 views

Exist measurable functions $f_n$ with $\limsup_{n\to\infty} f_n (x) = \infty$ but $\lim_{n\to\infty} \int f_n = 0$?

Are there measurable functions $f_n: [0,1] \to [0, \infty)$, $n \in \mathbb{N}$, with $$\limsup_{n\to\infty} f_n (x) = \infty \qquad \forall x \in [0,1]$$ but $$\lim_{n\to\infty} \int_{[0,1]} f_n (x) ...
0
votes
2answers
40 views

Does $\int _{\Omega} \left | f_{n} \right |^p \rightarrow \infty $ implies $\left | f_{n}(x) \right | \rightarrow \infty $ pointwise?

Let $f_{n}$ be sequence of Lebesgue integrable functions. $\Omega$ is open, bounded subset of $\mathbb{R}^n$. We know that if $f_n$ converges to some limit in $L^p$, then we have a subsequence $ ...
0
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1answer
48 views

Is Riemann–Lebesgue lemma valuble in $L2(\mathbb{R})$

If $f\in L_1$ on $\mathbb{R}$, that is to say, if the Lebesgue integral of $|f|$ is finite, then the Fourier transform of $f$ satisfies $$\hat{f}(z):= \int_{\mathbb{R}} f(x)e^{-izx} dx \rightarrow 0, ...
0
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1answer
51 views

Test whether $f$ is Lebesgue integrable or NOT

Let , $f[0,1]\to \mathbb R$ be defined by $\displaystyle f(x)=\begin{cases}\frac{1}{x}\sin \frac{1}{x} &\text{ if }x\not =0\\0 &\text{ if } x=0\end{cases}$. Show that $f$ is NOT Lebesgue ...
0
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1answer
9 views

Product Measure Spaces and Order of Integration

Let $(X,S,\mu)$ and $(Y,T,\nu)$ be two $\sigma$-finite measure spaces. Define $\phi_{A}(x)=\nu(A_{x})$ for $A_{x}=\{y \in Y: (x,y) \in A\}$ and $A \in S \otimes T$ (the $\sigma$-field generated by the ...
2
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2answers
55 views

$L^2$ bound on polynomial

I want to show the following theorem: Let $\frac{1}{1+|p|} \in L^2(\mathbb{R}^d)$ where $p$ is some polynomial. Then $\forall z \in \mathbb{R} \backslash A$ for $A:=\overline{\{p(x);x \in ...
1
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0answers
68 views

Show that for almost all $x$ in $[-1,1]$, the series $ \sum\limits_{n=1}^\infty \frac{1}{2^n \sqrt{|x-r_n|}} $ converges

Let $\{r_n\}$ be a sequence of real numbers in $[-1, 1]$, then show that for almost all $x$ in $[-1,1]$, the series $$ \sum_{n=1}^\infty \frac{1}{2^n \sqrt{|x-r_n|}} $$ converges. I am struggling ...
1
vote
2answers
100 views

Show that $f$ is NOT Lebesgue integrable.

Define $f:[0,1]\to \mathbb R$ by $f(x)=\begin{cases}(-1)^nn &\text{ if }\frac{1}{n+1}<x\le \frac{1}{n}\\0 &\text{ otherwise }\end{cases}$ Show that $f$ is NOT Lebesgue integrable in ...
0
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2answers
81 views

Does $\int_E f = 0$ implies $f = 0$ a.e.?

I know that for nonnegative $f$ and a measurable set $E$, $$\int_E f$$ (Lebesgue integral) is identically zero iff $f = 0$ a.e. How can I show this property holds for any arbitrary $f$ (not ...
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0answers
6 views

Vanishing of certain integrals over $\partial B(0,R)$

Why does integrals like $$ \int_{|y|=R}\frac{y_k^2-\frac{|y|^2}{n}}{|y|^{n+2}}\,d\sigma_n(y) $$ or $$ \int_{|y|=R}\frac{y_ky_j}{|y|^{n+2}}\,d\sigma_n(y) $$ vanish? Here $d\sigma_n(y)$ denotes the ...
0
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1answer
132 views

$\partial_{x_j}\partial_{x_i}\int_{\mathbb{R}^n}k(x,y)d\mu_y=\int_{\mathbb{R}^n}\partial_{x_j}\partial_{x_i}k(x,y)d\mu_y$ under some assumptions

Let $k:\mathcal{O}\times\mathbb{R}^n\to\mathbb{R}$, with $\mathcal{O}\subset\mathbb{R}^m$ open, be such that $\forall x\in\mathcal{O}\quad k(x,\cdot)\in L^1(\mathbb{R}^n) $, i.e. the function ...
0
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1answer
42 views

Differentiating $\int_{\mathbb{R}^3} \frac{g(y)}{\|y-x\|} d\mu_y$ under the integral

I know that if $f\colon \mathbb{R}^3 \to \mathbb{R}$ is integrable on any measurable, according to the usual $n$-dimensional Lebesgue measure $\mu_y$, and bounded subset of $\mathbb{R}^3$, and ...
1
vote
1answer
61 views

On the solution of Poisson equation

Given $f\in\mathcal{C}_{c}^2(\Bbb R^n)$ (i.e. twice real-differentiable with compact support) I'm in trouble with the following passage $$ ...
-2
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1answer
75 views

Help with Lebesgue integration [closed]

I want to solve integration of $\sin(x)$ from $0$ to $\pi$ ,with the help of measure ..
0
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2answers
56 views

Monotone increasing sequence in Lp convergent a.e.

I´m having trouble with the proof of the following theorem in measure theory: Consider the sequence $(f_n)_{\,n \in \Bbb N}\subset L^p(X)$ with $\;0\le f_n(x)\le f_{n+1}(x) \;\;\;\forall x\in ...
0
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0answers
17 views

approximation of nonnegative, non simple functions and integral

Let $f(x) = \begin{cases} \frac{1}{|x|} & \text{if } 0 < |x| < \\|x| & \text{if } |x| \geq 1 \\0 & \text{if } x = 0 \end{cases}$ The set $(E_n)^c$ is $\{x : f(x) \geq ...
1
vote
1answer
86 views

A differentiation under the integral sign.

I know that if $f\colon \mathbb{R}^n \to \mathbb{R}$ is integrable on any measurable, according to the usual $n$-dimensional Lebesgue measure $\mu_y$, and bounded subset of $\mathbb{R}^n$, and ...
1
vote
1answer
31 views

Lebesgue integral confusion

I am trying to understand Lebesgue integration of simple functions. If $(X,A,\mu)$ is a measure space, $E \in A$ and $\mu(E) = 0$ and $f(x) = \begin{cases} 0 & \text{if } x \in E \\ 1 ...
0
votes
1answer
33 views

$f^2g$ integrable then $fg$ integrable

Given a probability space $(\Omega, \mathscr{F}, P)$, fix measurable functions $f,g:X \to \mathbb{R}$ such that $f^2g$ is integrable. Is also $fg$ integrable?
1
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1answer
29 views

Lebesgue measure for Radon-Nikodym derivative

$\mathbb{P}_0$ is Lebesgue measure, $\mathbb{P}_1$ is the probability measure given by $\mathbb{P}_1([a,b])=\int_a^b (4\omega-1) \, d\mathbb{P}_0(w)$ $\Omega$- is the interval $[0,1]$. I need to find ...
1
vote
1answer
74 views

Differentation under the integral sign

Let I be an open subinterval of $\mathbb R$ and let $f:\mathbb R \to \mathbb R$ be a Borel measurable function such that $x \to e^{tx}f(x)$ is Lebesgue integrable for each t in I. Define $h: I \to ...
0
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1answer
54 views

Fourier transform of $\frac{1}{\|x\|} \chi_{B_1(0)}(x)$

Define $f: R^3 \rightarrow R, f(x) = \frac{1}{\|x\|} \chi_{B_1(0)}(x)$ (with $f(0) = 0$). I would like to calculate the fourier transform $g(\xi) = \int_{R^3} f(x) e^{-ix\cdot \xi}dx$. I tried ...
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0answers
32 views

Calculating $\iiint_{A}x^pdxdydz$ where $A=\{x^2+y^2+z^2<x^{\frac{1}{3}}\}$

I have a problem with the following task, I need to calculate this integral: $\iiint_{A}x^pdxdydz$ where $A$ is $$A=\{x^2+y^2+z^2<x^{\frac{1}{3}}\}$$ At first glance it looks like they want me to ...
1
vote
1answer
46 views

Symmetric matrix with integral entries and nonnegative determinant

I would like to propose a generalization of another question which I posed here yesterday. The main reason is the heuristic that if an inequality holds for the finite case, then an integral analogue ...
0
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0answers
36 views

In which $L^p$-Space is the following function?

I got a $C^\infty(\Omega)$ function $f$ with $f(g(x))\leq C(1+|g(x)|)^3$. In which $L^p$-Space ist $f(g(x))$ if $g(x)$ is in $L^6(\Omega)$? Where $\Omega\subset\mathbb{R}^d$ with $d=2,3$ is a ...
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0answers
25 views

Calculating 3-dimensional Volume of 4-dimensional Graph

Let $D = \{(x,y,z) \in R^3 | |x|<|z|^2, |y|<|z|, 0<z<1 \}$ and $f: D\rightarrow R, f(x,y,z)=2x+2y+z^3$. I would like to calculate the 3-dimensional volume of $G := \{(x, f(x)) | x \in ...
0
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1answer
32 views

Uniform convergence theorem fails when |E| =$\infty$

Uniform Convergence Theorem: Let $f_k \in L(E)$ for $k=1,2,...$, and let ${f_k}$ converge uniformly to $f$ on $E$, $|E|<\infty$. Then $f\in L(E)$ and $\int_E f_k \rightarrow \int_E f $. I need to ...
1
vote
1answer
25 views

Close to diagonal implies derivative is close to 1?

A function $f:[a,b]\to\mathbb R$ is absolutely continuous if it has a derivative $f'$ almost everywhere with respect to Lebesgue measure, $f'$ is Lebesgue integrable, and $f(x)=f(a)+\int_a^x f'(t)dt$ ...
3
votes
1answer
29 views

Is $(T,\omega) \mapsto \int_0^T f(t,\omega)\ dt$ measurable?

Let $(\Omega, \mathcal{A}, P)$ be a probability space. Denote by $\mathcal{B}$ the Borel space on the real line and denote by $\mathcal{B}_{[0, \infty)}$ the Borel space on the interval $[0, \infty)$. ...
0
votes
0answers
42 views

Lebesgue integral measurable in the variable not of integration?

Let $X$, respectively $Y$, be a space endowed with the $\sigma$-additive complete measure $\mu_x$, respectively $\mu_y$. If $f: X\times Y\to\mathbb{R}$ is $\mu_x\otimes\mu_y$-measurable$^1$ and ...
0
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0answers
35 views

Summability of $\frac{1}{\|\boldsymbol{r}-\boldsymbol{l}\|^2}$ on $V\times[a,b]$

Let $\boldsymbol{r}$ represent a point of $\mathbb{R}^3$ with two components fixed and one, say $r_k$, free to vary on $[a,b]$, and let $V\subset\mathbb{R}^3$ be a measurable, according to the ...
0
votes
1answer
18 views

Convergence of ${f_n}$ of measurable function

For a sequence {$ {f_n} $} of measurable function in a set $ A $ of finite measures, show that $\lim_{n\rightarrow \infty} \int_A \frac{|f_n|}{1+|f_n|}dm=0$ iff {$f_n$} converges to zero in ...
2
votes
0answers
35 views

Differentiability under the integral sign of absolutely continuous $f(-,x)$

I read that if $f:[a,b]\times X\mapsto\mathbb{R}$, where $X$ is a space endowed with measure$\mu_x$ and we define in $[a,b]\subset\mathbb{R}$ the usual Lebesgue linear measure $\mu_t$, ...
2
votes
2answers
31 views

Find $p$ that a function will be in $L^p_\text{loc}$

Let the function $G(x)=\ln|x|$ defined in $\mathbb{R}^2 \setminus \{0\}$. How can we prove that $\nabla G \in L^p_\text{loc} ((]0,1[)^2)$ when $p<2$? I calculate $\nabla G = ...
0
votes
0answers
34 views

collection of even, Lebesque integratable functions

A function $f:\mathbb{R} \to \mathbb{C} $ is even if $f(x)=f(-x)$ for all x. Let $E$ be the collection of all even Lebesque integrable functions on $\mathbb{R}$. for $f\in E$ let ...