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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30
votes
2answers
9k views

Limit of $L^p$ norm

Could someone help me prove that given a finite measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$, ...
34
votes
2answers
10k views

$L^p$ and $L^q$ space inclusion

Let $(X, \mathcal B, m)$ be a measure space. For $1 \leq p < q \leq \infty$, under what condition is it true that $L^q(X, \mathcal B, m) \subset L^p(X, \mathcal B, m)$ and what is a counterexample ...
11
votes
2answers
3k views

Generalisation of Dominated Convergence Theorem

Wikipedia claims, if $\sigma$-finite the Dominated convergence theorem is still true when pointwise convergence is replaced by convergence in measure, does anyone know where to find a proof of this? ...
3
votes
2answers
311 views

Continuity of $L^1$ functions with respect to translation

Let $f\in L^1$, consider the map $t\mapsto f_t=f(x-t)$, then how can one show that $t\mapsto f_t$ is continuous? More explicitly one wants to show that $\lim_{h\to 0}|f_{t+h}-f_t|_{L^1}=0$. I tried to ...
1
vote
3answers
462 views

$L^p$-norm of a non-negative measurable function

Can I ask a homework question here? Let $f$ be measurable and non-negative in $\mathbb R^d.$ Using Fubini's theorem, show that for $1 \leq p \lt \infty,$ $$\lVert f\rVert^p_p = ...
32
votes
9answers
7k views

Lebesgue integral basics

I'm having trouble finding a good explanation of the Lebesgue integral. As per the definition, it is the expectation of a random variable. Then how does it model the area under the curve? Let's take ...
0
votes
1answer
132 views

Derivative of Fourier transform: $F[f]'=F[-ixf(x)]$

Let us define the Fourier transform of the Lebesgue-summable function $f\in L_1(\mathbb{R},\mu_x)$ as $F[f](\lambda)=\int_{\mathbb{R}}f(x) e^{-i\lambda x} d\mu_x$, where $\mu_x$ is the Lebesgue linear ...
4
votes
2answers
74 views

$\int_{\mathbb{R}}f(x)e^{-ixz}d\mu_x$ analytic for $f\in L_1$

Let $f\in L_1(-\infty,\infty)$ be a Lebesgue-summable function on $\mathbb{R}$ and let $x\mapsto e^{\delta|x|}f(x)$ also be Lebesgue-summable on all the real line. I have added the condition that ...
2
votes
1answer
96 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 ...
7
votes
2answers
3k views

Is Dirichlet function Riemann integrable?

"Dirichlet function" is meant to be the characteristic function of rational numbers on $[a,b]\subset\mathbb{R}$. On one hand, a function on $[a,b]$ is Riemann integrable if and only if it is bounded ...
3
votes
2answers
786 views

pointwise convergence and boundedness in norm imply weak convergence

I am contemplating over the following exercise (in which $E=[0,1]$): Let $f_n$ be a sequence of functions in $L^p(E)$, $1<p<\infty$, which converge almost everywhere to a function $f$ in ...
5
votes
1answer
1k views

Integral vanishes on all intervals implies the function is a.e. zero

I am having trouble with the following problem: $f:\mathbb{R}\to \mathbb{R}$ is a measurable function such that for all $a$: $$\int_{[0,a]}f\,dm=0.$$ Prove that $f=0$ for $m$ almost every $x$ ...
3
votes
1answer
74 views

For a distribution function $F(x)$ and constant $a$, integral of $F(x + a) - F(x)$ is $a$.

For any distribution function and any $a \geq 0$, $\int_{-\infty}^{\infty} (F(x+a)-F(x))dx = a$. In this case, "distribution function" means a right continuous function F with $F(-\infty) = 0$, ...
1
vote
3answers
135 views

Show that $\lim _{r \to 0} \|T_rf−f\|_{L_p} =0.$

I am having a hard time with the following real analysis qual problem. Any help would be awesome. Thanks. Suppose that $f \in L^p(\mathbb{R}),1\leq p< + \infty.$ Let $T_r(f)(t)=f(t−r).$ Show ...
1
vote
1answer
307 views

Riemann-Lebesgue lemma

How can I prove the following result? Let $([-1,1],M,m)$ a measure space, where $m$ is the Lebesgue measure in $[-1,1]$. If $f$ is Lebesgue integrable, then ...
10
votes
3answers
2k views

Can a function that has uncountable many points of discontinuity be integrable?

First of all, I would like to show you how we defined Riemann-integrals and Lebesgue-integrals to make sure that we are talking about the same: Riemann-intregrability Let $f:\mathbb{R} \rightarrow ...
1
vote
1answer
2k views

Lebesgue Integral but not a Riemann integral

Is it possible for a function to be a Lebesgue integral, but not a Riemann integral? After the comments below I realize my question was not a good one. Thank you. This is my edited version: Let $f$ ...
8
votes
2answers
569 views

Reinventing The Wheel - Part 1: The Riemann Integral [closed]

Preface The core of any notion of integral is some sort of weighted sum: $$\sum b\mu(A)$$ Depending on wether the domain or range is decomposed these split into Riemann and Lebesgue type ones: ...
6
votes
1answer
162 views

Showing that a function is in $L^1$

I need to prove the following statement or find a counter-example: Let $u\in L^1\cap C^2$ with $u''\in L^1$. Then $u'\in L^1$. Unfortunately, I have no idea how to prove or disprove it, since the ...
2
votes
2answers
3k views

Showing that $1/x$ is NOT Lebesgue Integrable on $(0,1]$

I aim to show that $\int_{(0,1]} 1/x = \infty$. My original idea was to find a sequence of simple functions $\{ \phi_n \}$ s.t $\lim\limits_{n \rightarrow \infty}\int \phi_n = \infty$. Here is a ...
22
votes
2answers
555 views

Category Theory and Lebesgue Integration.

I'm wondering if there's any Category Theory floating around in the theory of Lebesgue Integration. To avoid things becoming too broad, let's keep this focused on the basics. Here's how I see the ...
6
votes
1answer
348 views

If a function is Riemann integrable, then it is Lebesgue integrable and 2 integrals are the same?

Is is true that if a function is Riemann integrable, then it is Lebesgue integrable with the same value? If it's true, how to prove it? If it's false, what is a counterexample?
13
votes
1answer
229 views

If $f$ is Lebesgue integrable on $[0,2]$ and $\int_E fdx=0$ for all measurable set E such that $m(E)=\pi/2$. Prove or disprove that $f=0$ a.e.

Let $f$ be a Lebesgue integrable function on $[0,2]$. If $\int_E fdx=0$ for all measurable set $E$, such that $m(E)=\pi/2$. Is $f=0$ a.e. Prove or disprove I could not figure out anything. Can a ...
1
vote
2answers
511 views

Integrability of Derivative of a Continuous Function

Let $f$ be continuous on $[a,b]$ and has finite derivative a.e. on $[a,b]$. Let $f_n(x)=n[f(x+1/n)-f(x)] $ s.t. $f_n$ be uniformly integrable on $[a,b]$. I want to show : $f'$ is Lebesgue integrable. ...
9
votes
1answer
185 views

Lebesgue space and weak Lebesgue space

Let $1\le p<\infty$. We define the weak Lebesgue space $wL^p(\mathbb{R}^d)$ as the set of all measurable functions $f$ on $\mathbb{R}^d$ such that \begin{equation} \|f\|_{wL^p}=\sup_{\gamma>0} ...
6
votes
2answers
3k views

Does Riemann integrable imply Lebesgue integrable?

Suppose a definite integral exists in the Riemann sense. Does that mean the integral exists as a Lebesgue integral, and do we get the same result either way? ------- BTW: I have a MS in Electrical ...
5
votes
2answers
1k views

Riemann-integrable (improperly) but not Lebesgue-integrable

Show that for any $\alpha>1$ the function $$ g_{\alpha}(x):=\sin(x^{\alpha}), x\geq 0 $$ is improperly Riemann-integrable but not Lebesgue-integrable on $\mathbb{R}_+$. Could ...
3
votes
2answers
126 views

How do we prove $\int_I\int_x^1\frac{1}{t}f(t)\text{ dt}\text{ dx}=\int_If(x)\text{ dx}$

Let $f:\mathbb{R}\to\mathbb{R}$ be Borel-measurable and Lebesgue-integrable over $I:=(0,1)$. Further, let $\;\;\;\;\;\;\;\;\;\;g : I\to \mathbb{R}\;,\;\;\; \displaystyle x ...
6
votes
3answers
172 views

a characterization of $L^p$ space

The following question should be part of the questions I recently asked here Prove or disprove a claim related to $L^p$ space If $g \in L^p(\Omega, \lambda)$ where $\Omega$ is a bounded subset of ...
5
votes
1answer
300 views

Yet another definition of Lebesgue integral

Let $[a, b]$ be a finite interval of the real line. A partition $P$ of $[a, b]$ is a finite sequence of numbers of the form $a = t_0 < t_1 <\cdots < t_{k-1} < t_k = b$ Let $(X, \mu)$ be ...
4
votes
1answer
72 views

Prove g is Lebesgue intergrable

Let $f$ be Lebesgue integrable on $(0, 1)$. For $0 < x < 1$ define g(x) = $\int_x^1t^{-1}f(t)dt$ Prove that $g$ is Lebesgue integrable on $(0, 1)$. $\int^1_0g(x)dx=\int^1_0f(x)dx.$ I am not ...
4
votes
1answer
118 views

Show $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $

Need to prove $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $ and $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{y}{1+y^2}e^{-ay} dy =0 $ Can ...
3
votes
3answers
79 views

If $\int_A f d\mu =\int_A g d\mu$ for $\forall A\in \mathcal A$ and $\mu$ is $\sigma$-finite then $f=g$ a.e

Functions $f,g$ are nonnegative on $(X,\mathcal A, \mu)$. If $\int_A f d\mu =\int_A g d\mu$ for $\forall A\in \mathcal A$ and $\mu$ is $\sigma$-finite then $f=g$ a.e Can I discard the condition ...
3
votes
1answer
64 views

Show that $\lim_{t \to 0} \int_{\mathbb{R}^d}|f(x)-f(x-t)|dx = 0$

Let $f \in L^1(\mathbb{R}^d)$. Show that $\lim_{t \to 0} \int_{\mathbb{R}^d}|f(x)-f(x-t)|dx = 0$. What I want to do is bound $|f(x)-f(x-t)|$ above by something and then use the Lebesgue Dominated ...
3
votes
1answer
145 views

Exercise on measure theory

Let $X\neq \emptyset$ and $f:X \rightarrow [0, \infty]$ not identical infinity. Set $$ \sum_{x \in X} f(x)= \sup \left\{ \sum_{x \in F}f(x), F \subseteq X, F \mbox{ finite} \right\}.$$ $(i)$ Show ...
1
vote
1answer
67 views

A continuous $L^1$ function $f : \mathbb{R}\rightarrow \mathbb{R}$..

For a continuous function $f :\mathbb{R}\rightarrow \mathbb{R}$ satisfying $$\int_{\mathbb{R}}|f(x)|dx<\infty$$ and for some $\alpha >0$ let $d_f(\alpha)$ be tthe lebesgue measure of the set ...
5
votes
1answer
379 views

Norm of Fredholm integral operator equals norm of its kernel?

Let $T_k(f)(s):=\int_0^1 k(s,t) f(t) dt $, where $k \in L^2([0,1]^2)$ and $f \in L^2([0,1])$. Then it was fairly easy to see that $||T_k|| \le ||k||_{L^2}$, but now I was wondering how to show that ...
4
votes
1answer
202 views

Properties of special rectangle (measure)

Let $I$ be a special rectangle in $\mathbb{R}^n$, and denote $\lambda(A)$ the measure of $A$. Prove that the following conditions are equivalent: a) $\lambda(I)=0$ b) $I^{\circ}=\emptyset$ (i.e., ...
4
votes
1answer
418 views

Extension of Fatou's lemma

let $X$ be a finite measure space and $\{f_n\}$ be a sequence of integrable functions, $f_n \rightarrow f\text{ a.e.}$ on $ X$. I want to show if (1) holds, then (2) holds too. $$\lim_{n \rightarrow ...
3
votes
3answers
111 views

How to prove that $L^p [0,1]$ isn't induced by an inner product? for $p\neq 2$

I'd like to know how could i prove that $L^p [0,1]$ isn't induced by an inner product? (For $p\neq 2$, including $p=\inf$). It is clear to me that i would need to find two functions $f$, $g$ in $L^p$ ...
3
votes
2answers
197 views

Characterisation of one-dimensional Sobolev space

I've got some doubts proving that $$H^1_0((a,b))=\{u\in AC([a,b]): u'\in L^2 \text{ and } u(a)=u(b)=0\}:=X.$$Let $$\mathcal A=\{v\in C^2([a,b]):v(a)=v(b)=0\}.$$ $H^1_0((a,b))\subseteq X.$ Let ...
3
votes
1answer
102 views

Find a non-negative function on [0,1] such that $t\cdot m(\{x:f(x) \geq t\}) \to 0$ that is not Lebesgue Integrable

Problem: Find a non-negative function $f$ on $[0,1]$ such that $$\lim_{t\to\infty} t\cdot m(\{x : f(x) \geq t\}) = 0,$$ but $f$ is not integrable, where $m$ is Lebesgue measure. My Attempt: Let ...
3
votes
1answer
88 views

Show $\sqrt{1 + (\int_{\Omega} h d\mu)^2} \leq \int_{\Omega} \sqrt{1+h^2} d\mu$

in preparation for an exam I wanted to show that for $\mu(\Omega)=1$ and $h:\Omega \rightarrow [0,\infty]$ measurable the following inequality holds: $\sqrt{1 + (\int_{\Omega} h d\mu)^2} \leq ...
2
votes
1answer
39 views

Questions of an exercise in Lebesgue integral

I'm doing exercises related to Lebesgue integral and get stuck by two of them. I can't figure out what do some steps in solutions mean. Some definitions probably will be used: Definition of ...
2
votes
1answer
68 views

Measure Spaces: Uniform & Integral Convergence

Given a measure space $\Omega$. Consider a sequence of measurable functions $f_n$ Suppose it converges pointwise: $f_n\to f$ Can one find increasing subsets with uniform convergence: ...
2
votes
3answers
289 views

Integral change that I don't understand

If $f(t)$ is a Probability density function of a positive RV. $\int_0^\infty\int_x^{\infty}f(t)dtdx$ Using fubini theorem should become $\int_0^\infty\int_0^{t}f(t)dxdt$ But why? Surely the answer ...
2
votes
0answers
225 views

Vector-Lattices and “Approximating $\mathscr{L^1(\mathbb{R}^k)}$”.

In this question I asked whether $\mathscr{L}^1(\mathbb{R}^k)$ forms a category in any way. It was concluded that indeed it does not. I thought to myself, "well, could we at least approximate the ...
2
votes
2answers
163 views

Integral of Schwartz function over probability measure

Let $X$ be a set, $\mathcal F$ a $\sigma$-field of subsets of $X$, and $\mu$ a probability measure on $X$. Given random variables $f,g\colon X\rightarrow\mathbb{R}$ such that ...
2
votes
3answers
108 views

summable square function implies…?

I have difficulty to demonstrate this: $$ \int_{-\infty}^{\infty}|f(x)|^2dx<\infty~~~\text{(summable square function)}$$ then, $$\lim_{|x|\rightarrow\infty }f(x)=0$$ thank you.
2
votes
1answer
296 views

Bounded measurable function and integral with charcteristic function

I have been struggling with the following for quite some time now. If anyone can give me some help, it will be much appreciated: Let $f$ bounded, measurable and $E$ be a set of finite measure. Let $A ...