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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50
votes
2answers
17k 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 ...
43
votes
8answers
10k 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 ...
25
votes
7answers
2k views

How much do we really care about Riemann integration compared to Lebesgue integration?

Let me ask right at the start: what is Riemann integration really used for? As far as I'm aware, we use Lebesgue integration in: probability theory theory of PDE's Fourier transforms and really, ...
24
votes
2answers
735 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 ...
20
votes
6answers
675 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
629 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 thought ...
16
votes
3answers
1k views

How to decide whether Lebesgue integral or Riemann integral?

Very often I feel very uncomfortable in dealing with integrals, since I am wondering whether the given integral is meant as a (improper) Riemann integral or Lebegue integral? For instance, the Gamma ...
16
votes
2answers
2k 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 ...
16
votes
1answer
326 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 ...
12
votes
2answers
4k 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? ...
12
votes
1answer
151 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 L^1(\...
12
votes
1answer
166 views

$L^2(\mathbb{R})$ sequence such that $\sum_{n=1}^{\infty}\int_{\mathbb{R}}f_n(x)g(x)d\mu(x)=0$

I am currently studying for an analysis qualifying exam, and this problem has been bothering me: Suppose we have a sequence $\{f_n\}$ in $L^2(\mathbb{R})$ such that $\sum_{n=1}^{\infty}||f_n||_2^2<\...
12
votes
2answers
498 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 ...
12
votes
1answer
660 views

Why is Lebesgue-Stieltjes a generalization of Riemann-Stieltjes? Moreover, is there an example where Lebesgue-Stieltjes is useful

I certainly have a question, but i don't know what the best title should be. Please edit the title if there is a better one :) And I believe, to get a better answer, it would be good to explain ...
11
votes
1answer
1k 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 $t \to f(x, t)$ ...
10
votes
2answers
5k 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 ...
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 \...
10
votes
1answer
190 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 \...
10
votes
1answer
1k 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?
10
votes
1answer
154 views

Lebesgue measure and characterisation of function $\Phi$ [Rudin-Real&Complex]

Let $m$ be Lebesgue measure on $[0,1]$ and define $||f||_p$ with respect to $m$ as usual. What are all functions $\Phi$ on $[0, \infty)$ such that the relation $$ \Phi( \lim_{p\to\ 0}||f||_p)= \int_{...
10
votes
1answer
827 views

The integral of a characteristic function with respect to a product measure.

Problem: Let $ (X,\mathcal{A},\mu) $ and $ (Y,\mathcal{B},\nu) $ be measure spaces, where $ X = Y $ is the interval $ [0,1] $, $ \mathcal{A} = \mathcal{B} $ is the collection of Borel ...
10
votes
2answers
6k views

Is there a solution manual for Royden fourth edition?

I bought the fourth edition of Royden Real Analysis, this book is awesome and is quite different of third edition that has less excersices. I have the solution manual for the third edition. Is there ...
9
votes
5answers
859 views

Evaluating Integrals using Lebesgue Integration

Suppose we are to evaluate: $$I = \int_{0}^{1} f(x) dx$$ Where $$f(x)=\begin{cases}1 \space \text{if} \space x\space \text{is rational}, & \newline 0 \space \text{if} \space x \space \text{is ...
9
votes
2answers
378 views

Lebesgue integration of simple functions

Define $f : [0,1] \to \Bbb R$ by $f(x) := 0$ if $x$ is rational, and $f(x) := d^2$ if $x$ is irrational, where $d$ is the first nonzero digit in the decimal expansion of $x$. Show that $\int_{[0,1]...
9
votes
2answers
407 views

Why is the undergraph definition of Lebesgue integral so rare?

So in Pugh's Real Mathematical Analysis, the initial definition of the Lebesgue integral is as the Lebesgue measure of the undergraph of the function (where the function is nonnegative, with the usual ...
9
votes
1answer
436 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 ...
9
votes
1answer
447 views

Definitions of Lebesgue integral

I know the definition, from A.N. Kolmogorov and S.V. Fomin's Элементы теории функций и функционального анализа, of Lebesgue integral of measurable function $f:X\to \mathbb{C}$ on $X,\mu(X)<\infty$ ...
9
votes
1answer
281 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} \...
9
votes
1answer
131 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 = \lim_{n\to\infty}\...
9
votes
1answer
207 views

Significance and applications of the Riesz Representation Theorem in locally compact Hausdorff spaces

Can anyone tell me the signification of Theorem $2.14$ (The Riesz Representation Theorem in locally compact Hausdorff spaces), page $40, 41$ in Rudin - Real and Complex Analysis? And some applications ...
9
votes
1answer
137 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 ...
9
votes
0answers
231 views

$\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$

Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded Lipschitz domain and $u$ is a measurable function. A sufficient condition for the integral $\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^...
8
votes
3answers
928 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 ...
8
votes
1answer
6k views

Bounded convergence theorem

Suppose that $f_n$ is a sequence of measurable functions that are all bounded by M, supported on a set E of finite measure, and $f_n(x)\to f(x)$ a.e. x as $n\to \infty$. Then f is measurable, bounded, ...
8
votes
6answers
263 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
1answer
839 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 ...
8
votes
3answers
762 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
2answers
138 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} ...
8
votes
2answers
79 views

Show that the set $\{x \in \mathbb{R}| \lim_{n \to \infty} \sin(a_n x) \mbox{ exists}\}$ has zero measure

$a_n$ is a sequence of real numbers such that $a_n \to +\infty$. Show that the set $E = \{x \in \mathbb{R}| \lim_{n \to \infty} \sin(a_n x) \mbox{ exists}\}$ has zero (Lebesgue) measure. The hint for ...
8
votes
1answer
2k 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
468 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 ...
8
votes
1answer
431 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 $f$....
8
votes
1answer
1k 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
1answer
90 views

Intuition behind proof of bounded convergence theorem in Stein-Shakarchi

Theorem 1.4 (Bounded convergence theorem) Suppose that $\{f_n\}$ is a sequence of measurable functions that are all bounded by $M$, are supported on a set $E$ of finite measure, and $f_n(x) \to f(x)$ ...
8
votes
2answers
605 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: $$\{...
8
votes
0answers
387 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 \...
8
votes
1answer
1k 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 $f \in \mathfrak R(\alpha)$ ...
8
votes
2answers
560 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 ...
7
votes
2answers
432 views

Integrating over the naturals. What does it mean?

Let $F$ be the power set of $\Bbb{N}$ and consider the measurable space $(\Bbb{N}, F)$. Then what does it mean to take the integral with respect to the measure $\mu(A) = \sum_{a \in A} \frac{1}{a}$. ...
7
votes
3answers
1k views

General condition that Riemann and Lebesgue integrals are the same

I'd like to know that when Riemann integral and Lebesgue integral are the same in general. I know that a bounded Riemann integrable function on a closed interval is Lebesgue integrable and two ...