Questions tagged [lebesgue-integral]

For questions about integration, where the theory is based on measures. It is almost always used together with the tag [measure-theory], and its aim is to specify questions about integrals, not only properties of the measure.

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$L^p$ 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 ...
Confused's user avatar
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120 votes
8 answers
51k 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 ...
user957's user avatar
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120 votes
3 answers
73k views

Does convergence in $L^p$ imply convergence almost everywhere?

If I know $\| f_n - f \|_{L^p(\mathbb{R})} \to 0$ as $n \to \infty$, do I know that $\lim_{n \to \infty}f_n(x) = f(x)$ for almost every $x$?
187239's user avatar
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57 votes
3 answers
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Functions that are Riemann integrable but not Lebesgue integrable

I know there are functions which are Riemann integrable but not Lebesgue integrable, for instance, $$\int_{\mathbb{R}} \frac{\sin(x)}{x} \mathrm{d}x$$ Is Riemann integrable and it is easily shown ...
user438666's user avatar
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50 votes
8 answers
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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, ...
Ennar's user avatar
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48 votes
8 answers
6k views

Why do we restrict the definition of Lebesgue Integrability?

The function $f(x) = \sin(x)/x$ is Riemann Integrable from $0$ to $\infty$, but it is not Lebesgue Integrable on that same interval. (Note, it is not absolutely Riemann Integrable.) Why is it we ...
Rachel's user avatar
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48 votes
6 answers
7k views

What is the intuition behind Chebyshev's Inequality in Measure Theory

Chebyshev's Inequality Let $f$ be a nonnegative measurable function on $E .$ Then for any $\lambda>0$, $$ m\{x \in E \mid f(x) \geq \lambda\} \leq \frac{1}{\lambda} \cdot \int_{E} f. $$ What ...
Bill's user avatar
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40 votes
2 answers
3k 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 ...
Shaun's user avatar
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33 votes
4 answers
37k 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 ...
Vladimir's user avatar
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32 votes
4 answers
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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? ...
anegligibleperson's user avatar
32 votes
3 answers
41k views

What does it mean to be an $L^1$ function?

I am struggling to understand what the space $L^1$ is, and what it means for a function to be $L^1$. A friend told me that a function $f$ is $L^1$ if $\int_\mathbb{R} |f|$ is finite. It is $L^2$ if $(...
fourier1234's user avatar
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29 votes
2 answers
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Is there a solution manual for Royden fourth edition?

I bought the fourth edition of Royden Real Analysis, this book is awesome and it's quite different from the third edition, which has less exercises. I have the solution manual for the third edition. ...
28 votes
1 answer
2k views

Why is the Monotone Convergence Theorem more famous than it's stronger cousin?

I am reading Stein & Shakarchi. On page 62 we have the Monotone Convergence Theorem: Suppose $\{f_n\}$ is a sequence of non-negative measurable functions with $f_n\nearrow f.$ Then $\...
Ovi's user avatar
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28 votes
2 answers
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The Lebesgue integral of an integrable function is continuous

Let $a\in \mathbb{R}$ be fixed, and $f:\mathbb{R}\rightarrow \mathbb{R}$ a Lebesgue integrable function. Define $F:\mathbb{R}\rightarrow \mathbb{R}$ by $$F(x)=\int_a^xf(t)\mathrm{d}t$$ for all $t\in \...
Ducky's user avatar
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24 votes
5 answers
2k 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 ...
asmeurer's user avatar
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24 votes
0 answers
775 views

When is $\int_0^1 \int_0^1 \frac{f(x) - f(y)}{x-y} \, \text{d} x \, \text{d} y = 2 \int_0^1 f(t) \log\left(\frac{t}{1-t}\right) \, \mathrm{d} t$?

Double integrals of this type sometimes appear when using differentiation under the integral sign with respect to two variables. Therefore, I am interested in reducing them to (simpler) single ...
ComplexYetTrivial's user avatar
22 votes
6 answers
20k views

If $\int_0^x f \ dm$ is zero everywhere then $f$ is zero almost everywhere

I have been thinking on and off about a problem for some time now. It is inspired by an exam problem which I solved but I wanted to find an alternative solution. The object was to prove that some ...
Johan's user avatar
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22 votes
3 answers
1k 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 ...
Romeo's user avatar
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22 votes
3 answers
7k 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 \...
Martin Thoma's user avatar
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22 votes
3 answers
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Why is expectation defined by $\int xf(x)dx$?

I recently found out that the expectation of a random variable $X$ in a probability space $(\Omega, \mathcal F, \mathbb P)$, $\mathbb E(X)$, is just the term used in probability theory for the ...
John Gowers's user avatar
22 votes
2 answers
11k views

Differentiation under the integral sign for Lebesgue integrable derivative

The problem is the following: Let $a,b,c,d \in \mathbb R$ be given such that $a<b$ and $c<d$. Suppose $f : [a,b] \times [c,d] \to \mathbb R$ is a function such that $\partial_1 f: [a,b] \times [...
Sam's user avatar
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22 votes
3 answers
766 views

Is there a set $A \subset [0,1]$ such that $\int_{A \times A^\text{c}} \frac{\mathrm{d} x \, \mathrm{d} y}{\lvert x - y\vert}=\infty$?

The above question came up when I was trying to find a counterexample related to this problem. Clearly, the integral of $(x,y) \mapsto \lvert x-y \rvert^{-1}$ over $[0,1]^2$ is divergent. When ...
ComplexYetTrivial's user avatar
22 votes
1 answer
5k views

A rigorous meaning of "induced measure"?

In my readings I often come across terms like "induced measure" or "induced Lebesgue measure". For example: $$\int_{\mathbb{B}^n}u\frac{\partial v}{\partial x_j}\;dx = \int_{\mathbb{S}^{n-1}}uv\...
Patch's user avatar
  • 4,245
21 votes
3 answers
4k views

Intuitively understanding Fatou's lemma

I learnt Fatou's lemma a while ago. I am able to prove it and use it. I know examples showing that the inequality may be strict. But I don't really have an intuitive way to understand it. Any good ...
pqpq's user avatar
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21 votes
1 answer
11k 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?
unknown's user avatar
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21 votes
2 answers
5k 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 ...
Bachmaninoff's user avatar
  • 2,251
20 votes
3 answers
9k 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 ...
user156043's user avatar
20 votes
2 answers
1k views

Practicality of the Lebesgue integral

I am getting pretty frustrated with the Lebesgue integral mainly because it seems highly impractical to calculate anything non-trivial. Whenever I look for a concrete calculation all I see are ...
Amara's user avatar
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20 votes
1 answer
3k 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 ...
John. p's user avatar
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19 votes
3 answers
2k 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 Lebesgue integral? For instance, the ...
Hasti Musti's user avatar
19 votes
2 answers
3k views

Various $p$-adic integrals

Here are three (possibly different) definitions of $p$-adic integrals that I have encountered during my self-studies. First of all, here is what Vladimirov, Volovich and Zelenov write at the beginning ...
Santiago's user avatar
  • 1,586
19 votes
2 answers
2k views

Show that if the integral of function with compact support on straight line is zero, then $f$ is zero almost everywhere

I want to prove that that given $f:R^2 \rightarrow R$ which is continuous with compact support s.t the integral of $f$ for every straight line $l$ is zero ($\int f(l(t))\mathrm{d}t=0$) then $f$ is ...
user2323232's user avatar
18 votes
2 answers
11k views

Why is the Monotone Convergence Theorem restricted to a nonnegative function sequence?

Monotone Convergence Theorem for general measure: Let $(X,\Sigma,\mu)$ be a measure space. Let $f_1, f_2, ...$ be a pointwise non-decreasing sequence of $[0, \infty]$-valued $\Sigma-$measurable ...
Bear and bunny's user avatar
18 votes
3 answers
9k views

A function that is Lebesgue integrable but not measurable (not absurd obviously)

I think: A function $f$, as long as it is measurable, though Lebesgue integrable or not, always has Lebesgue integral on any domain $E$. However Royden & Fitzpatrick’s book "Real Analysis" (4th ...
Thang's user avatar
  • 827
18 votes
2 answers
5k views

Lebesgue integral of a positive function on a set of positive measure

Let $E$ be a subset of $\Bbb R$ with positive Lebesgue measure, $\lambda(E)>0$. Let $f$ be a function from $\Bbb R$ to $\Bbb R$ which is positive on $E$, that is $f(x)>0$ for all $x\in E$. Is ...
User's user avatar
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18 votes
3 answers
12k views

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

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 $x$ (...
rondo9's user avatar
  • 1,906
18 votes
1 answer
3k 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 $t \to f(x, t)$ ...
Diav's user avatar
  • 333
17 votes
2 answers
2k views

Does the graph of a measurable function always have zero measure?

Question: Let $(X,\mathscr{M},\mu)$ be a measure space and $f\colon X \to [0,+\infty[$ be a measurable function. If ${\rm gr}(f) \doteq \{ (x,f(x)) \mid x \in X \}$, then does it always satisfy $$(\mu ...
Ivo Terek's user avatar
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16 votes
2 answers
8k views

Measure Convergence Version of Lebesgue Dominated Convergence Theorem

I want to prove that LDCT(Lebesgue Dominated Convergence Theorem) continues to hold if I replace the hypothesis $f_n \to f$ (convergence pointwise) with $f_n\to f$ (convergence in measure): $$\int ...
89085731's user avatar
  • 7,614
16 votes
4 answers
2k views

On the horizontal integration of the Lebesgue integral

I'm studying Lebesgue integral and its difference with respect to the Riemann one. I'm reading that the key difference (at least graphically speaking) is that the first slices the function ...
Marco's user avatar
  • 771
16 votes
1 answer
2k 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 ...
JHF's user avatar
  • 11k
16 votes
2 answers
3k views

Space $\mathcal{L}^p(X, \Sigma, \mu)$ is separable iff $(\Sigma, \rho_\Delta)$ is separable

Let's consider the space $\mathcal{L}^p(X, \Sigma, \mu)$ of all functions $f\colon X \to \mathbb{R}$ (or $\mathbb{C}$) for which: $$ \int\limits_X|f|^p \mu(dx) < \infty. $$ Here $X$ is a metric ...
Glinka's user avatar
  • 3,192
16 votes
1 answer
765 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 ...
seriously divergent's user avatar
16 votes
1 answer
4k views

Is there a general theory of the "improper" Lebesgue integral?

I like the Lebesgue integral a lot more than the other alternatives, because of its connections to measure theory; it's a way of thinking about integration that's as "liberated" from the structure of ...
goblin GONE's user avatar
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16 votes
1 answer
2k views

Are continuous functions strongly measurable?

Measure theory is still quite new to me, and I'm a bit confused about the following. Suppose we have a continuous function $f: I \rightarrow X$, where $I \subset \mathbb{R}$ is a closed interval and $...
ScroogeMcDuck's user avatar
15 votes
2 answers
6k 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 ...
D_S's user avatar
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15 votes
4 answers
8k views

Prove that $f$ is integrable if and only if $\sum^\infty_{n=1} \mu(\{x \in X : f(x) \ge n\}) < \infty$

Problem statement: Suppose that $\mu$ is a finite measure. Prove that a measurable, non-negative function $f$ is integrable if and only if $\sum^\infty_{n=1} \mu(\{x \in X : f(x) \ge n\}) < \infty$....
poppy3345's user avatar
  • 2,182
15 votes
3 answers
4k 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 $L^p$ ...
user avatar
15 votes
1 answer
6k views

Suppose $1\le p < r < q < \infty$. Prove that $L^p\cap L^q \subset L^r$.

Suppose $1\le p < r < q < \infty$. Prove that $L^p\cap L^q \subset L^r$. So suppose $f\in L^p\cap L^q$. Then both $\int |f|^p d\mu$ and $\int|f|^q d\mu$ exist. For each $x$ in the domain of $...
Christmas Bunny's user avatar
15 votes
1 answer
3k 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 ...
Jake Casey's user avatar
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