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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3
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1answer
51 views

Increasing functions on $\mathbb{R}$

If $F$ is increasing on $\mathbb{R}$ then show that $F(b)-F(a)\geq \int_a^b F'(t)dt$. My work: Since $F$ is increasing on $\mathbb{R}$, $F'$ exists a.e. on $\mathbb{R}$. So $F'$ is integrable on ...
0
votes
2answers
30 views

If $f_n \rightarrow 0$ and $\int \sup f_1, … , f_n \leq M$ then $\int f_n \rightarrow 0$

Let $f_n$ be a sequence of nonnegative measurable functions which converge to $0$. If there exists an $M$ such that $$\int \sup f_1, ... , f_n \leq M$$ for all $n$, then $\lim \int f_n = 0$. Could ...
4
votes
2answers
45 views

Prove the following inequality: $\int_{(a,b)}f\ d\lambda\cdot\int_{(a,b)}\frac{1}{f}d\lambda≥(b-a)^2$

Assignment: Let $-\infty < a < b < \infty$ and $f: (a,b) \rightarrow (0,\infty)$ be measurable, such that $f$ and $\frac{1}{f}$ are Lebesgue integrable. Prove the following inequality: ...
1
vote
1answer
23 views

Existence everywhere of integrand in Fubini's theorem

Let $f\in L(A,\mu_x\otimes\mu_y)$ be a summable function on $A\subset X\times Y$ where $(X\times Y,\mu_x\otimes\mu_y)$ is the product of measure spaces $(X,\mu_x)$ and $(Y,\mu_y)$. Then Fubini's ...
1
vote
1answer
26 views

Measurability of inner integral $x \mapsto \int f(x,y)\, d\mu(y)$

Let $\psi$ be defined by$$\psi(s):=\int_{[a,b]}K(s,t)\varphi(t)d\mu_t$$ where $\varphi\in L_2[a,b]$ and $K\in L_2([a,b]^2)$. Kolmogorov-Fomin's proves the belonging of $\psi$ to $L_2[a,b]$ by showing ...
1
vote
1answer
27 views

an inequality on $L_p$ and $l_2$

Let $\{{f_i}\}$ be a countable or finite collection of good functions (e.g. Schwartz functions on $\mathbb{R}$). Let $1<p\le2$. Is it true that ...
1
vote
0answers
16 views

Definition of Measure regular

Book's, Real and complex analysis, Walter Rudin. I am somewhat confused. My question is: "In other words, we are looking at $L^p$, where $\mu$ is Lebesgue measure on $[0,2\pi]$(or on $T$), ...
2
votes
1answer
51 views

estimate of infinite norm by $(p,q)$ norms

Let $p$ and $q$ be conjugate exponents, i.e. $\frac{1}{p}+\frac{1}{q}=1$. Prove or disprove: $$ \|f\|_\infty^2\le\|f\|_p\|f'\|_q $$ I think this is true. I tried to prove it using integration by ...
2
votes
2answers
44 views

Confused about substitution in Stiltjes integral

Suppose we have an integral $$ \int_{-a}^{a} \sin (x) \nu(dx), $$ where $\nu$ is a finite measure with $\nu(-A)=\nu(A), A \in \sigma(\mathbb{R})$ and $x>0$. Then we have $$ \int_{-a}^{a} \sin (x) ...
1
vote
1answer
24 views

What does it mean to be absolutely integrable on $\mathbb{R}$ and what are the steps to show that something is absolutely integrable?

I just have a quick question. What does it mean to be absolutely integrable on $\mathbb{R}$ and what are the steps to show that something is absolutely integrable? For example what if we wanted to ...
3
votes
1answer
55 views

Evaluation of Lebesgue Integral using Convergence Theorems

Using convergence theorems, I am trying to compute the value of $$ \lim_{n\to\infty}\int_a^\infty \frac n{1+n^2x^2}\,\mathbb{d}x $$ for $a \in \mathbb{R}$, and with respect to the Lebesgue measure. ...
0
votes
1answer
25 views

Bochner Integral: Approximability

Disclaimer This thread is related to: Bochner Integral: Integrability It is meant to record. See: Answer own Question It is written as jeopardy. Have fun! :) Problem Given a measure space ...
1
vote
1answer
26 views

Showing that $||\hat{f}||_{\infty} \leq ||f||_1$ in $L^1$

Let $f \in L^1(\mathbb{R}^n)$ then $\hat{f} \in L^{\infty}(\mathbb{R}^n)$ and $||\hat{f}||_{\infty} \leq ||f||_1$ How do you prove this or where can I find a proof of this fact?
2
votes
0answers
34 views

Proving translational invariance of Lebesgue integral

I am asked to show that the Lebesgue integral is invariant under translations. Specifically, Let $(\mathbb{R}, \Sigma, \mu)$ be a measure space, and for any $f:\mathbb{R}\rightarrow\mathbb{R}$ ...
1
vote
1answer
22 views

Degenerate Hilbert-Schmidt operators

Let us define a Hilbert Schmidt operator $A:L_2[a,b]\to L_2[a,b]$ by $$A\varphi:=\int_{[a,b]} K(s,t)\varphi(t)d\mu_t$$where $\mu_t$ is the linear Lebesgue measure. A degenerate case is represented by ...
1
vote
2answers
85 views

Amann & Escher Integral vs. Lebesgue Integral

In the textbook the authors define the integral via cauchy sequences of simple functions: $$S_n\to F:\quad\int F\mathrm{d}\mu:=\lim_n\int ...
0
votes
0answers
24 views

Derivative of $g(t) = \int_0^t u(s)$ for $u \in L^2(0,T)$.

Let $u \in L^2(0,T)$. Consider $g(t) = \int_0^t u(s)\;ds$. Is it true that $g'(t) = u(t)$ a.e? How would I show this? Does some stronger statement hold?
0
votes
1answer
32 views

Measurability of function defined by an integral

Let $A$ be a Hilbert-Schmidt operator defined on $L_2[a,b]$ by $$A(\varphi):=\int_{[a,b]}K(s,t)\varphi(t)d\mu_t$$where $K\in L_2([a,b]^2)$. The fact that $A(\varphi)\in L_2[a,b]$ is showed in the ...
-1
votes
1answer
19 views

Properties of function on $L_p$ spaces

Given $L_p$ space with the lebesgue measure on $\mathbb{R}^n$ and the function $f(x) = |x|^{-\alpha}$ if $|x| < 1$ $f(x) = 0$ if $|x| \geq 1$ I need to show that $f \in L_p$ if and only if ...
0
votes
1answer
40 views

Help in proving $f \circ \phi \in \mathcal L^1(\lambda) \iff \int_0^{\infty} \frac {f(x)}{\sqrt x} \lambda (dx) < \infty$

Consider the measure space $(\mathbb R, \mathcal B(\mathbb R), \lambda)$ and let $\phi: \mathbb R \rightarrow \mathbb R$ be given by $\phi(x) = x^2$. I want to show that for $f \in \mathcal ...
3
votes
1answer
61 views

Question on $L_p$ spaces involving $\lambda^n$-measure on $\mathbb{R}^n$

Q/ Consider $L_p=L_p(\lambda^n)$ with the Lebesgue measure on $\mathbb{R}^n$ and $1\leq p<\infty$. Let $f_0=|x|^{-\alpha}$ for $|x|<1$ and $0$ otherwise. Show $f_0\in L_p$ iff $p\alpha < n$. ...
1
vote
1answer
52 views

Proving uniform convergence of an integral-defined function on compact sets

If $f$ is a compactly supported smooth (infinitely differentiable) function into $[0, 1]$ such that $\int f(x)dx = 1$, $g$ is a continuous function, and $f_\epsilon(x) = ...
8
votes
2answers
97 views

How can using a different definition for the integral be useful?

It's often said that the Lebesgue integral is superior to the Riemann integral because it satisfies nicer properties, for instance things like $$\lim_{n\to\infty} \int f_n = \int \lim_{n\to\infty} ...
0
votes
0answers
27 views

Investigate the existance of the following integrals (in a measure theory context)

I've been asked to investigate the existence and equivalence of the these integrals: $$\int^0_1\int^0_1f(x,y)d\lambda(x)d\lambda(y)\text{ and }$$ $$\int^0_1\int^0_1f(x,y)d\lambda(y)d\lambda(x)$$ (yes ...
1
vote
1answer
57 views

Question on $L_p$ spaces

Consider $L_p = L_p(\lambda^n)$ with the Lebesque measure on $\mathbb{R}^n$ and $1 \leq p < \infty$. Let $f_0(x) = |x|^{-\alpha}$ if $|x| < 1, f_{0}(x) = 0$ for $|x| \geq 1$. Show that: $f_{0} ...
1
vote
0answers
31 views

Fourier transform of distribution

Let $f\in S_{\infty}$ be a Schwartz function and let us define a linear functional,for any $\varphi\in S_{\infty}$, $S_{\infty}\to\mathbb{C}$, $\varphi\mapsto (f,\varphi)$ ...
2
votes
0answers
26 views

Integrating a function of measures

I've been reading John Baez's series of posts on Information Geometry. I'm currently on part 6... Midway through the post he discusses Radon-Nikodym derivatives: The formula for information gain ...
1
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0answers
31 views

Commutativity of Lebesgue-Stieltjes convolution

Let $F,G$ be non-decreasing real functions of bounded variation on $\mathbb{R}$ and $\mu_F$ the Lebesgue-Stieltjes measure defined by it. Kolmogorov-Fomin's says (p. 452 here) that we can commute the ...
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0answers
35 views

Can someone check my answer to a measure theory question on existence and equality of three integrals.

I have been told to investigate the existence and equality of the integrals; $\int_{[0,1]^2} f\;d\lambda^2$, $\int_0^1\int_0^1 f\;d\lambda(x)d\lambda(y)$ and $\int_0^1\int_0^1 ...
1
vote
0answers
28 views

Inequality related to Lebesgue-Stieltjes convolution

Let $F_1:\mathbb{R}\to\mathbb{C}$ and $F_2:\mathbb{R}\to\mathbb{C}$ be functions of bounded variation on $\mathbb{R}$. In order to prove the bounded variation of their convolution defined as ...
2
votes
2answers
50 views

Approximate $f$ by simple functions

Consider the measure space $([0,1)^2, \mathcal{B}([0,1)^2, \lambda)$ where $\lambda$ is the Lebesgue measure on $[0,1)^2$. Put $f:[0,1)^2 \to \mathbb{R}, \,\,\, f(x,y) = x + 2y$. I would like to ...
3
votes
1answer
72 views

A question on Fubini's Theorem

We have been asked to investigate whether $\int_{[0,1]^2} f \, d\lambda^2$ exists and if so whether it is equal to $\int_0^1 (\int_0^1f(x,y) \, d\lambda(x)) \, d\lambda(y)$ and $\int_0^1 (\int_0^1 ...
3
votes
0answers
33 views

Reversing an “inverse Fourier transform”

Let $g$ be the Fourier transform of an unknown function $y\in L_1(-\infty,\infty)$:$$g(\lambda)=\int_{\mathbb{R}}y(x)e^{-i\lambda x}d\mu_x$$Let $f$ be defined as ...
3
votes
0answers
37 views

Convergence in Measure, Different Definitions

Let $(X, \mu)$ be a measure space, $E \subseteq X$ measurable, and $f_n$ a sequence of measurable functions on $E$. If $f$ is another function on $E$, I have seen two definitions for what it means ...
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0answers
30 views

Lebesgue integrability of $x^a \sin(x)$ on $[0,\infty[$

Why is $x^a \sin(x)$ Lebesgue-integrable on $[0,\infty[$? It's obviously measurable, so why $\int |f|<\infty$... You can divide the integral by the period of sine, but I don't know how to cope with ...
0
votes
0answers
27 views

Inversion formula for $\int_{\mathbb{R}}f(x)e^{-izx}dx$

Let $f:\mathbb{R}\to\mathbb{C}$ be a measurable function such that$$\forall x\ge 0\quad|f(x)|<Ce^{\gamma_0 x}$$$$\forall x<0\quad f(x)=0$$I must specify that all the integrals I am going to ...
1
vote
1answer
18 views

calculate $\lim_{n\to\infty}\int_{[0,\infty)} \exp(-x)\sin(nx)\,\mathrm{d}\mathcal{L}^1(x)$

We've had the following Lebesgue-integral given: $$\int_{[0,\infty)} \exp(-x)\sin(nx)\,\mathrm{d}\mathcal{L}^1(x)$$ How can you show the convergence for $n\rightarrow\infty$? We've tried to use ...
2
votes
1answer
62 views

If $\int_0^{x} g \leq \int_0^x f$ and $\phi$ is nonincreasing then $\int_0^{\infty} \phi g \leq \int_0^\infty \phi f$

Let $f, g$ be measurable real-valued functions on $[0, \infty)$, with $$\int_0^{x} g \leq \int_0^x f$$ for each $x$. Show that if $\phi: [0, \infty) \rightarrow [0, \infty)$ is nonincreasing, then ...
1
vote
1answer
43 views

If $\int_0^1 f(y)\sin(xy) dy = 0$ for every $x$, then $f = 0$ almost everywhere.

Can someone please give me a hint on this question, I have no idea where to start. Let $f \in L^p$ for some $1 \leq \infty$. Assume for all $x \in [0,1]$ that $$\int_0^1 f(y)\sin(xy) dy = 0$$ Show ...
0
votes
1answer
33 views

Prove $\lim_{n \rightarrow \infty} n \cdot \lambda(\{ f > n \}) = 0$. Where $\lambda$ is the lebesgue measure.

Suppose $f$ is integrable over $E$. And the assumptions are as given above. Then, currently, I have from Chebyshev's inequality, $$ \lambda( \{ f > n \} ) \leq \frac{1}{n}\int_E |f| $$ Thus, ...
2
votes
1answer
27 views

example of a sequence $f_n$, $n=1,2…$ of integrable functions converging to $f$ s.t limit of integral of $f_n$ does not exist

Is there an example of a sequence of of functions $f_n$ converging to a function $f$ such that $f_n$, $n=1,2...$ are integrable and nonegative and their integral over a measurable set $A$ is less than ...
0
votes
0answers
37 views

Lebesgue integration of $f(x)=\frac{1}{x}$ where $x\in[0,3]$

We have the function $f(x)=\infty$ if $x=0$ and $f(x)=\frac{1}{x}$ if $x$ otherwise. So, in this two values of function, I made simple approximation of $f(x)$ by the help of simple function : ...
0
votes
1answer
30 views

Convergence in L^p, Cauchy in L infinity

If $u_n$ is a convergent sequence in $L^p$ with $u_n \to u$, and $u_n$ is convergent is $L^\infty$, is it true that the limit in $L^\infty$ must be $u$? Is it true if $u_n$ are all test functions, ...
2
votes
1answer
71 views

$f \in L^1$, but $f \not\in L^p$ for all $p > 1$

"Find an $f \in [0,1]$ such that $f \in L^1$ but $f \not\in L^p$ for any $p > 1$." I've thought about doing something like $$f(x) = \frac{1}{x}$$ where $|f|^p = \frac{1}{x^p}$ doesn't converge ...
4
votes
0answers
57 views

How is Riemann–Stieltjes Integration insufficient for developing modern probability theory?

If we consider Riemann–Stieltjes integration then it can perfectly account for mixed probability distribution (a continuous R.V with some point mass). So why would we still need Lebesgue Integration ...
2
votes
0answers
18 views

Measurability of $f:X\times Y\to\mathbb{K}$ and $f(-,y):X\to\mathbb{K}$

Let $(X,\mu_x)$ and $(Y,\mu_y)$ be two measure spaces endowed with $\sigma$-additive compete measures $\mu_x$ and $\mu_y$, respectively. Let $\mu:=\mu_x\otimes\mu_y$ be the Lebesgue extension of ...
0
votes
1answer
21 views

$ \int_{\mathbb{R}^n} f^p dx$ for $p>0$ and measurable $f$

Let $f: \mathbb{R}^n \rightarrow \mathbb{\overline{R}} $ be non-negative and (Borel)-measurable and $p>0$. Then: $$ \int_{\mathbb{R}^n} f^p dx = p \int_{0}^{\infty} t^{p-1} ...
-1
votes
1answer
30 views

Uniform convergence and integrability

If $(f_n)_{n \in \Bbb N}$ converges to $f$ uniformly and each $f_n$ integrable would it imply $f$ is integrable and $$\lim_{n \to \infty}\int f_n = \int f$$ In case each $f_n$ is nonnegative ...
4
votes
1answer
65 views

Fourier transform inversion formula for $f\in L_1(\mathbb{R}^n)$ and Dini condition

Let us define the Dini condition for a function $f\in L_1(-\infty,\infty)$, i.e. Lebesgue summable on $\mathbb{R}$, as Given an $x\in\mathbb{R}$ there is a $\delta>0$ such that the Lebesgue ...
0
votes
0answers
23 views

Orthonormal bases in L^p

Let $(X,\mu), (Y,\nu)$ be $\sigma$-finite measure spaces. Suppose $\phi_1$ and $\phi_2$ are total sets in $L^p(\mu)$ and $L^p(\nu)$, respectively. How do I show that $\{f\otimes g: f\in \phi_1, g \in ...