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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23
votes
2answers
6k 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 ...
18
votes
6answers
510 views

Is there a fundamental reason that $\int_b^a = -\int_a^b$

Is there a fundamental reason that switching the order of the limits in an integral results in the negative, i.e., $$\int_b^af(x)\,dx = -\int_a^bf(x)\,dx?$$ As far as I can tell, this is just chosen ...
18
votes
3answers
514 views

If $\int_{\mathbb R^2} \frac{\vert f(x)-f(y)\vert}{\vert x-y\vert^2}dxdy<+\infty$ then $f$ is a.e. constant

Let $f \in L^1(\mathbb R)$. If $$ \int_\mathbb R \int_\mathbb R \frac{\vert f(x)-f(y)\vert}{\vert x-y\vert^2}dxdy<+\infty $$ then $f$ is a.e. constant. I do not know how to begin. I ...
17
votes
2answers
326 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 ...
14
votes
2answers
4k views

Limit of $L^p$ norm

Could someone help me prove that given a measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^{\infty}$ and some $L^{q}$, ...
14
votes
2answers
317 views

A snappy proof of Fatou's lemma

I'm studying basic real analysis and stumbled across three big results (Fatou's lemma, Lebesgue's Dominated Convergence theorem, and the Monotone Convergence theorem) in the theory of Lebesgue ...
12
votes
3answers
186 views

$\int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}=\left|\det\left({\pi}^{-1}A\right)\right|^{-1/2}$ for every positive definite matrix $A$

Let $A$ be a positive definite real $n\times n$ matrix. How can I prove that $$ \int_{\mathbb{R}^n}e^{-\langle Ax,x\rangle}\text{d}x=\left|\,\det\left(\pi^{-1}{A}\right)\right|^{-1/2}\!, $$ where ...
12
votes
1answer
124 views

Does $f_n \to 0$ in $L^1(\mathbb R^2)$ imply that $f_{n_k}(x,\cdot)\to 0$ in $L^1(\mathbb R)$ for almost every $x \in \mathbb R$?

I would like to know what you think about this question. It is a "self-posed" question: I formulated it while I was doing an exercise. Suppose you have $(f_n)_{n\ \in \mathbb N}\subset ...
12
votes
2answers
225 views

Topology of convergence in measure

Currently I am doing some measure theory (on $X=[0,1]$ with the Borel-Sigma algebra and the Lebesgue measure), and I am looking at sets $A \subset L^p$, such that for all $q \in (0,p)$, the topologies ...
11
votes
1answer
119 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 ...
9
votes
3answers
1k 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 ...
9
votes
1answer
135 views

$\int_X |f_n - f| \,dm \leq \frac{1}{n^2}$ for all $n \geq 1$ $\implies$ $f_n \rightarrow f$ a.e.

Let $(X, M, m)$ be an arbitrary measure space. Let $f_n, f \in L^1_m(X)$. Assume that $$\int_X |f_n - f| \, dm \leq \frac{1}{n^2} \text{ for all }n \geq 1. $$ Then I want to show that $f_n ...
9
votes
1answer
110 views

Evaluating $\sum_{n=0}^{\infty}\ \int_{\pi/4}^{\pi/3}\sin^{n}x (1-\sin x)^2 dx$ using a convergence theorem

$$\sum_{n=0}^{\infty}\ \int_{\pi/4}^{\pi/3}\sin^{n}x (1-\sin x)^2 dx$$ Let $g_n = \sin^{n}x (1-\sin x)^2$ $g_n$ is a sequence of measurable functions and $g_n \ge 0$ so applying the Beppo Levi ...
8
votes
6answers
217 views

give an example of a function that is integrable in $\mathbb R $ and $\lim_{ x\to \infty}f(x)\neq0$

i did a search for such function but didn't found anything useful/complete ! , like this : Integrable function $f$ on $\mathbb R$ does not imply that limit $f(x)$ is zero is there any function that ...
8
votes
3answers
416 views

How do I prove $f=0$ almost everywhere?

During one of the problems in Rudin I was asked to show $f=0$ a.e. Here $f$ satisfies this condition: $$f(x)=\frac{1}{x}\int^{x}_{0}f(t)dt$$ almost everywhere and is in $L^{p}(0,\infty)$. So constant ...
8
votes
1answer
230 views

Holder's inequality for infinite products

In analysis, Holder's inequality says that if we have a sequence $p_1, p_2, \ldots, p_n$ of real numbers in $[1,\infty]$ such that $\sum_{i=1}^n \frac{1}{p_i} = \frac{1}{r}$, and a sequence of ...
8
votes
1answer
827 views

I want to understand uniform integrability in terms of Lebesgue integration

According to my Real Analysis textbook, a family $\scr{F}$ of measurable functions on $E$ is said to be uniformly integrable over $E$ provided for each $\epsilon$ $>$ $0$, there is a $\delta$ ...
8
votes
1answer
113 views

A dominated convergence theorem applied to $e$ number definition

I want to show that: $$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n=\sum_{k=0}^\infty \frac{1}{k!}.$$ By the binomial theorem $$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n = ...
8
votes
1answer
261 views

Limit of measurable functions in finite measure space

Let $(X,\mathcal{M},\mu)$ be a measure space with $\mu(X)<\infty$. Let $f_n$ be a sequence of measurable real-valued functions such that $f_n$ converges pointwise a.e. to a real-valued function ...
8
votes
1answer
529 views

How to find a measurable but not integrable function or a positive integrable function?

For an arbitrary interval $I$, how can we find a positive on $I$ integrable function? And how does one construct a measurable but not integrable function. If not all measurable functions are ...
8
votes
2answers
402 views

Banach space valued integration (Riemann type)

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: ...
8
votes
0answers
487 views

Difference of differentiation under integral sign between Lebesgue and Riemann

Here is a consequence of Lebesgue dominated convergence theorem on differentiation under integral sign. Function $f(x, t)$ is differentiable at $x_0$ for almost all $t \in A$, and $t \to f(x, t)$ ...
7
votes
2answers
54 views

Integral limit of $\sin(x/n)f(x)$

For any $f\in L^1[0,\pi]$, evaluate $n\to \infty \int^\pi_0 n$sin$(x/n)f(x)dx$ My idea is, $n$sin$(x/n)f(x)\to xf(x)$ and it seems that it is increasing sequence. I am not able to show it is ...
7
votes
2answers
197 views

Measure theory questions

i. If $1 < p < \infty$ and $E = \{f_a, a \in A\}$ set of measurable functions of $\mathbb{R}$ and $\sup_{a \in A} ||f_a||_p < \infty$, I want to show that for $ 0 < q < p$, $\lim ...
7
votes
1answer
119 views

Dense subspace of $L^{2}[0,1]$

I know that $C[0,1]$ is dense in $L^{2}[0,1]$ but is $\{f\in C^{2}[0,1]:f(0)=f(1)=0\}$ dense in $L^{2}[0,1]$?
7
votes
3answers
218 views

Convergence of functions in $L^p$

Let $\{f_k\} \subset L^2(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is a bounded domain and suppose that $f_k \to f$ in $L^2(\Omega)$. Now if $a \geq 1$ is some constant, is it possible to say ...
7
votes
1answer
202 views

$L^{p}$ functions from Rudin Exercises 3.5

I am attempting a question from Rudin's "Real and Complex Analysis" Chapter 3 question 5. I shall summarise the question as below: Suppose that $f$ is a complex measurable function on $X$, $\mu$ a ...
7
votes
0answers
265 views

Egorov's theorem for this Lebesgue integral

I want to prove Egorov's theorem using this Lebesgue integral defined by the upper integral $$\int^*f:=\left\{\int h ; h \ge f \text{ and h upper-continuous }\right\}$$ $$\int_*f:=\left\{\int h ; h ...
6
votes
3answers
111 views

Finding the integral of $\frac{x}{e^x + 1}$ [duplicate]

I've having some difficulty with finding this integral: $$ \int_0 ^{\infty} \frac{x}{e^x + 1}$$ Now usually I would use the monotone convergence theorem to write (using geometric series): $$f_n (x) ...
6
votes
3answers
419 views

Prove that $f=0$ almost everywhere.

Let $f$ be a Lebesgue integrable function on $[0,1]$ such that, for any $0 \leq a < b \leq 1$, $$\biggl|\int^b_a f(x)\,dx\,\biggr| \leq (b-a)^2\,.$$ Prove that $f=0$ almost everywhere. I would ...
6
votes
2answers
114 views

Partial integration for lebesgue integrable functions

I want to show the following: Let be two Lebesgue integrable functions given: $f,g:[a,b] \rightarrow \mathbb R$. We define the functions: $$F,G: [a,b] \rightarrow \mathbb R : F(x)=\int_{[a,x]}^ \! ...
6
votes
2answers
170 views

Evaluating $\int_{0}^{\pi/2}\log\left(a^2\cos^2\left(x\right)+b^2\sin^2\left(x\right)\right)\,{\rm d}x$

I am trying to evaluate the integral below by differentiating through the integral. Let $ F(a,b) :=\displaystyle\int_{0}^{\pi/2}\log\left(a^2\cos^2\left(x\right)+b^2\sin^2\left(x\right)\right)\,{\rm ...
6
votes
2answers
204 views

Invariance of the Lebesgue integral.

Problem Let $f\in L^1(\mathbb{R})$. Show that $\int_{\mathbb{R}}f(x)dx=\int_{\mathbb{R}}f(x-\frac{1}{x})dx$. Discussion I know the Lebesgue integral is translation invariant (as the Lebesgue measure ...
6
votes
2answers
107 views

Which integration theory to use?

As far as I know the Lebesgue integral generalises the Riemann integral. One key ingredient to this generalisation is that the Lebesgue integral partitions the range, not the domain as in the Riemann ...
6
votes
1answer
63 views

Consequence of Cauchy Schwarz in $\mathscr{L}^2$?

If $f,g\in \mathscr{L}^2$, then $\|fg\|_1\leq\|f\|_2\|g\|_2$. My textbook says that this is a consequence of Cauchy-Schwarz inequality. How so? Cauchy-Schwarz says that $|\langle ...
6
votes
1answer
278 views

Motivation behind introduction of measure theory

Is the motivation behind the introduction of measure theory the Lebesgue integral? In order to evaluate such an integral we need the length of each of the horizontal strip of width $h$. I have a ...
6
votes
1answer
146 views

Homogenous measure on the positive real halfline

Define a measure $\mu\not=0$ on positive real number $\Bbb R_{>0}$ such that for any measurable set $E\subset\Bbb R_{>0}$ and $a\in \Bbb R_{>0} $, we have $\mu(aE)= \mu(E)$, where ...
6
votes
2answers
279 views

How to apply Borel-Cantelli Lemma?

Assume that we are given a sequence of continuous functions $f_n(x)$ on $[0,1]$. How to show the existence of a sequence $a_n$ and a set $A$ with $\mu(A^c)=0$ so that $$ \lim_{ n\to \infty} ...
6
votes
1answer
457 views

Riemann-Stieltjes integrability criterion

I am currently reading through chapter 11 of Rudin's Principles of Mathematical Analysis, and I'm trying to solve problem 7: Find a necessary and sufficient condition that $f \in \mathfrak R(\alpha)$ ...
6
votes
0answers
65 views

Various integration theories

Could anyone briefly explain, or point me towards a resource explaining, the main differences between the main integration theories, namely: Riemann Integration Riemann-Stieltjes Integration ...
6
votes
2answers
323 views

constructing a sequence of simple functions with Lebesgue measure approaching the riemann integral

Let $\lambda$ denote the Lebesgue measure on the Borel sets of [0,1]. Let $f: [0,1] \rightarrow \mathbb{R}$ be continuous. I know that the Riemann integral $I:=\int_{0}^{1} f(x)dx$ exists. I also know ...
5
votes
3answers
535 views

Prove that function is not Lebesgue integrable

Prove that function $f(x,y)=\dfrac{1}{x^2+y^2}$ is not Lebesgue integrable on $A=(0,1]\times(0,1]$. To my knowledge the fastest way to do it is to use Fubini's theorem. From what I would get: ...
5
votes
4answers
761 views

Proof of Riemann-Lebesgue lemma

I read a book, and this mention to the following lemma of Rieman-Lebesgue type. Lemma. Let $-\infty<a<b<\infty$ and $f(x,y):[a,b]^2\to\mathbb R$ be an integrable and nonnegative function. ...
5
votes
2answers
206 views

Are integrable, essentially bounded functions in L^p?

Given an arbitrary measure space (of possibly infinite measure), if $f \in L^1 \cap L^\infty$, then by Hölder's inequality, $f^2 \in L^1$, so $f \in L^2$. Intuition suggests that $f \in L^p$ even for ...
5
votes
2answers
2k 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? ...
5
votes
2answers
191 views

How to prove $\int_0^\infty e^{-x}\frac{\sin^2 x}{x}\text{ dx}=\frac{\text{log }5}{4}$

I'm asked to prove $$\displaystyle\int_0^\infty e^{-x}\frac{\sin^2 x}{x}\text{ dx}=\frac{\text{log }5}{4}\tag{$\ast$}$$ by integration of $e^{-x}\text{sin}(2xy)$ over an suitable measurable ...
5
votes
2answers
1k 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 ...
5
votes
1answer
63 views

$\int_0^1f(x)dx = 2, \int_0^1g(x)dx = 1, \text{and} \int_0^1[f(x)]^2 dx ≤ C$ for some constant $C > 4.$

Suppose $f$ and $g$ are nonnegative measurable functions on the interval $[0,1],$ with the properties $$\int_0^1 f(x)\,dx = 2, \int_0^1g(x)\,dx = 1, \text{ and }\int_0^1[f(x)]^2 dx \le C$$ for some ...
5
votes
3answers
188 views

$ \int^{\infty}_0 |\frac{1}{(1+x)\sqrt x}|^p ~ \mathrm dx < \infty \implies p=?$

If $ f(x) = \frac{1}{(1+x)\sqrt x} $ how to find all $ p > 0 $ such that $$ \int^{\infty}_0 |f(x)|^p dx < \infty $$ The integral is with respect to lebesgue measure. Any solution or hints would ...
5
votes
2answers
442 views

How to prove that $\frac{\sin x}{x}$ is not Lebesgue integrable on $[0,+\infty]$?

How to prove that $\displaystyle\int_0^{+\infty}\left|\dfrac{\sin x}{x}\right| \, dx = +\infty$ ? Could any one give some hint ? Thanks.