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
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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3 votes
3 answers
2k 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. Suppose that $f \in L^p(\mathbb{R})$, where $1\leq p< + \infty$. Let $T_r(f)(t)=f(t−r)$. Show ...
kingkongdonutguy'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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10 votes
2 answers
5k 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 ...
Xuxu's user avatar
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32 votes
4 answers
13k 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? ...
anegligibleperson's user avatar
6 votes
3 answers
3k 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 = \int^{\infty}_{0}pt^{...
Park's user avatar
  • 109
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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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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18 votes
3 answers
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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
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7 votes
1 answer
2k views

Change of variable formula for a generic measure applied to classical change of variable formula.

I was reading this interesting post about changing the variables in an integral with a generic measure. I was wondering how this applies to the standard change of variable. In other words, $$\int_{F(\...
edamondo's user avatar
  • 1,287
7 votes
3 answers
1k views

Lebesgue Spaces and Integration by parts

Suppose there exists a Lebesgue Space, $L_1$ and functions functions $\phi$, $\phi'$, $f$, and $f'$ functions where $$\phi, \phi' \in L_1$$ By rule of integration by parts, $$uv|_a^b = \int_a^b udv + \...
Jeremy's user avatar
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3 votes
3 answers
798 views

Is $\frac{\vert x+y\vert}{1+\vert x+y\vert}\leq\frac{\vert x\vert}{1+\vert x\vert}+\frac{\vert y\vert}{1+\vert y\vert}$ true?

Is $\frac{\vert x+y\vert}{1+\vert x+y\vert}\leq\frac{\vert x\vert}{1+\vert x\vert}+\frac{\vert y\vert}{1+\vert y\vert}$ true for all $x,y\in\mathbb{R}$? If not, how can I prove that $\int\frac{\vert ...
Analysis's user avatar
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11 votes
4 answers
9k views

For every $\epsilon>0$ there exists $\delta>0$ such that $\int_A|f(x)|\mu(dx) < \epsilon$ whenever $\mu(A) < \delta$

Hello all mathematicians!! Again, I am struggling with solving the exercises in Lebesgue Integral for preparing the quiz. At this moment, I and my friend are handling this problem, but both of us ...
Block Jeong's user avatar
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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
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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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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6 votes
3 answers
888 views

Proof of the identity $\int_0^{+\infty}\frac{\sin(x)}{x^\alpha}dx=\frac{\Gamma(\alpha/2)\Gamma(1-\alpha/2)}{2\Gamma(\alpha)}$ for $\alpha\in (0,2)$.

Let $0<\alpha<2.$ Looking for a proof for the following: $$\int_0^{+\infty}\frac{\sin(x)}{x^\alpha}dx=\frac{\Gamma(\alpha/2)\Gamma(1-\alpha/2)}{2\Gamma(\alpha)}.$$ Any ideas?
Kurt.W.X's user avatar
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6 votes
1 answer
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The definition of Absolute Continuous function

Absolute Continuous functions require a finite sequence of subintervals of the domain. Can we replace this finite sequence with a countably infinite one in this definition? Please note that an ...
Matha Mota'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
  • 2,884
22 votes
3 answers
8k views

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
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
  • 1,173
12 votes
3 answers
9k 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 $L^p(...
mathreader's user avatar
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10 votes
2 answers
7k 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 ...
Ted Ersek's user avatar
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7 votes
1 answer
1k views

A differentiation under the integral sign

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a function Lebesgue summable on all $\mu$-measurable and bounded subsets of $\mathbb{R}^n$, where $\mu$ is the usual Lebesgue measure defined on $\mathbb{R}^n$, ...
Self-teaching worker's user avatar
4 votes
2 answers
9k 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 ...
Self-teaching worker's user avatar
4 votes
2 answers
184 views

$(\int f_1d\mu)^2+\cdots+(\int f_nd\mu)^2\leq(\int \sqrt{f_1^2+\cdots+f_n^2}d\mu)^2$

Let $(X, \mathfrak{B}, \mu)$ be a measurable space, possibly not $\sigma$-finite, and $f_1, \cdots, f_n \colon X\to (-\infty, +\infty)$ be integrable functions on $X$. Does $$(\int f_1d\mu)^2+\cdots+(\...
nessy's user avatar
  • 520
3 votes
1 answer
691 views

If $f\in L^1(\Bbb R,dx)$ then prove that for almost every $x\in\Bbb R$ $\lim\limits_{n\to \infty} f(nx) = 0.$

If $f\in L^1(\Bbb R,dx)$ Then prove that for almost every $x\in\Bbb R$ $$\lim_{n\to \infty} f(nx) = 0$$ Be aware this statement is different from the following: $$ f\in L^1(\Bbb R,dx)\implies \lim_{|...
Guy Fsone's user avatar
  • 23.9k
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
  • 705
14 votes
5 answers
7k views

Using Fatou's Lemmas in proving Scheffe's Lemma Part (ii)

Based on Williams' Probability w/ Martingales: Let $(S, \Sigma, \mu)$ be a measure space. Scheffe's Lemma Part (ii): Suppose $\{f_n\}_{n \in \mathbb{N}}, f \in \mathscr{L}^1 (S, \Sigma, \mu)$ and $\...
BCLC's user avatar
  • 13.5k
12 votes
2 answers
8k views

Riemann Integrable implies Lebesgue Integrable

We can say that a bounded function $f$ on a bounded interval $[a,b]$ is Riemann integrable if $$\sup \{\int_a^b\phi : \phi \le f, \phi \text{ step function} \} = \inf \{\int_a^b\phi dx : \phi \ge ...
SAS's user avatar
  • 1,066
11 votes
2 answers
11k views

Equivalent ideas of absolute continuity of measures

Wikipedia says that $\mu$ is absolutely continuous with respect to $\nu$, if $\nu(A)=0 \Rightarrow \mu(A)=0$. Okay, then I found another notion of absolute continuous measures: Let $||f||_1=1$ and $\...
user avatar
11 votes
2 answers
8k views

Prove that Lebesgue measurable set is the union of a Borel measurable set and a set of Lebesgue measure zero

Let $A$ be a Lebesgue measurable subset of $\Bbb R$. 1) Show that there exists a Borel measurable subset $B$ of $\Bbb R$ such that $A\subseteq B$ and such that $l^*(B\setminus A)=0$. 2) ...
user3029's user avatar
  • 303
7 votes
1 answer
519 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 $|\...
Lukas Lang's user avatar
6 votes
2 answers
1k 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 $f(...
dannum's user avatar
  • 2,519
0 votes
1 answer
309 views

Showing that $\{f_n \}$ converges to $f$ is equivalent to $\lim_{n\to \infty} \int_X \frac{|f_n(x)-f(x)|}{1+|f_n(x)-f(x)|}d\mu(x)=0$

Let $(X,\mathcal{M},\mu)$ be a measure space. We say that $\{f_n\}$ converges to $f$ in measure if, for any $\epsilon>0$ $$ \lim\limits_{n \to \infty} \mu\Big(\{x\in X: |f_n(x)-f(x)| \ge \epsilon ...
Jimmy Wang's user avatar
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
  • 9,774
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
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
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
13 votes
2 answers
22k 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 ...
user1770201's user avatar
  • 5,225
13 votes
2 answers
4k views

Definition of $L^\infty$

It seems like different authors use different definitions for the space $L^\infty$. For example wikipedia starts with bounded functions, and define the seminorm $\lVert f \rVert_\infty$, and take ...
Dilemian's user avatar
  • 1,015
11 votes
2 answers
795 views

Characterizing BV functions via the Stieltjes integral

Let $J=[a,b]$ be a compact interval and $f,g:J\rightarrow\mathbb{R}$ be two bounded functions. We say that $\int f dg$ exists (and equals $A$) if and only if the following condition holds: for any $\...
Dilemian's user avatar
  • 1,015
5 votes
2 answers
3k views

Generalization of Fatou's lemma for nonpositive but bounded measurable functions.

Let $(f_n)^{\infty}_{n=1}$ be a sequence of measurable (not-necessarily $\ge 0$). Let $g \gt 0$ be a measurable function with $\int g d\mu < \infty$ (integrable) such that $f_n\ge -g$ a.e. relative ...
alonso s's user avatar
  • 1,161
4 votes
2 answers
512 views

Show: $\int f\, d\mu=\int\limits_0^{\infty}\mu(\left\{x\in X: f(x)>t\right\})\, dt$

Let $f\colon X\to\overline{\mathbb{R}}_{\geq 0}$ be a measurable function. Show that $$ \int f\, d\mu=\int\limits_0^{\infty}\mu(\left\{x\in X: f(x)>t\right\})\, dt. $$ (The right integral ...
user avatar
2 votes
2 answers
14k views

Proving that $L^p \subset L^q$ when $1 \le q \le p$ [duplicate]

Let $(E,\mathcal{F},\mu)$ be a measure space such that $\mu(E)=1$ and let $L^p=L^p(E, \mathcal{F},\mu)$. Prove that $L^p \subset L^q\text{ if } 1 \le q \le p$. I let $f \in L^p$. Then $(\int_E |f|^pd\...
Bhavish Suarez's user avatar
1 vote
1 answer
757 views

Prove that $\int f\ d\lambda = \int_{a}^{b} f(x)\ dx,$ for any $f \in \mathcal R[a,b].$

Theorem $:$ Let $f : [a,b] \longrightarrow \Bbb R$ be a Riemann integrable function. Then $f \in L_1[a,b]$ and $$\int f\ d\lambda = \int_{a}^{b} f(x)\ dx.$$ The proof given in my book is as follows $:...
math maniac.'s user avatar
  • 2,013
1 vote
1 answer
655 views

Proof of countable additive property of Lebesgue Integrable functions

I am trying to prove the following Theorem: Let $\{A_1, A_2, \cdots \}$ be a countable disjoint collection of sets in $\mathbb R$ and let $S = \bigcup_{i=1}^\infty A_i$. Let $f$ be defined on $S$. (...
user avatar
50 votes
8 answers
7k 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, ...
Ennar's user avatar
  • 23.2k
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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