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.

1,399 questions with no upvoted or accepted 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 is quite different of third edition that has less excersices. I have the solution manual for the third edition. Is there ...
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345 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^...
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638 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 \...
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63 views

Relation between the measures of two sets defined via Lebesgue integration

Suppose $a : \mathbb R_+ \to \{-1,1\}$ is a measurable function. Let $X_0 =\frac12$. Consider a particle that moves on the $X-$axis as follows. $$X_t = X_0 + \int_0^t a_s ds$$ where the integral is a ...
9
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1answer
212 views

How is Riemann–Stieltjes Integration insufficient for developing modern probability theory?

If we consider Riemann–Stieltjes integration then it can perfectly account for mixed probability distribution (a continuous R.V with some point mass). So why would we still need Lebesgue Integration ...
9
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1answer
1k 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 $...
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129 views

Has the Hamiltonian Path Integral Been Made Rigorous?

It is well known that the Lagrangian formulation of the path integral has been made rigorous, via the Wiener measure and/or the Trottier product formula. I haven't seen mathematicians discuss the ...
7
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2answers
243 views

prove $ \sum_{n=1}^\infty \frac{1}{n^\alpha\sqrt{n |x-x_n|}} $ converges almost everywhere

Let $ \{x_n\}_{n=1}^\infty \subset \mathbb{R} $ be a sequence. Prove for $\alpha>1$ that $\sum_{n=1}^\infty \frac{1}{n^\alpha\sqrt{n|x-x_n|}}$ converges for almost every $x$ with regard to Lebesgue ...
7
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1answer
109 views

Integration/measure theory “paradox”?

I have encountered the following "paradox." Consider a dense countable subset of $\mathbb{R}$, e.g. $\mathbb{Q}$. Because the set is countable we may parametrise it by $\mathbb{Q} = \{ a_n \}_{n=1}^\...
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269 views

Spectacular failure of Lebesgue differentiation for rectangles

Let $\mathcal{R}$ be the set of rectangles in the plane and, given $f \in L^1$ let $$ f^*(x) = \sup_{x \in R \in \mathcal{R}} \frac{1}{ \lvert R \rvert} \int_R \lvert \, f \,\rvert $$ as defined in ...
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260 views

An a.e.-defined derivative which is not Lebesgue integrable on any interval?

If the derivative $f'$ exists everywhere then it is shown here that there exist intervals on which $f'$ is Lebesgue integrable. But perhaps there is a function $f$ such that $f'$ only exists almost ...
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3k views

Proving translational invariance of Lebesgue integral

I am asked to show that the Lebesgue integral is invariant under translations. Specifically, Let $(\mathbb{R}, \Sigma, \mu)$ be a measure space, and for any $f:\mathbb{R}\rightarrow\mathbb{R}$ ...
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237 views

Representation of Stochastic Integrals as Lebesgue/Bochner Integrals

Just as the Riemann–Stieltjes integral can be equivalently defined as a Lebesgue integral with the corresponding Lebesgue–Stieltjes measure, I am looking for the corresponding results for the ...
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Lebesgue Integration fundamental questions

My question involves the definition of the Lebesgue integral. Most colloquial definitions I've read follow (2), in that f*(t) is the "length" of one of the horizontal rectangles and dt is the height. ...
6
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1answer
98 views

When is this rearrangement theorem for integrals true?

Bernhard Riemann proved that if $(a_n)$ is a sequence in $\mathbb{R}$, then the sum of the infinite series $\Sigma_{n=1}^\infty a_n$ stays the same regardless of how you rearrange the terms if and ...
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112 views

Gagliardo–Nirenberg–Sobolev inequality for weighted Sobolev space with exponential weights

Consider the weighted $L^p_\omega(\mathbb{R}^d)$ space on $\mathbb{R}^d$ be the set of Lebesgue measurable functions such that $$\|f\|_{L^p_\omega}=\int_{\mathbb{R}^d}|f|^p\omega_\mu(x)\,dx< \...
6
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252 views

If $f \in L_p$ and $g \in L_\infty$, show that $fg \in L_p$.

If $f \in L_p, 1\le p\le \infty$, and $g \in L_\infty$, then the product $fg \in L_p$ and $\|fg\|_p \le \|f\|_p\|g\|_\infty$. I am still trying to show the first part, that $fg \in L_p$. What I have ...
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108 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 Lebesgue ...
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Compute the integral of $e^{-x}$

I am working on a problem in my past Qual. "Prove that $e^{-x}$ is Lebesgue integrable on $[0,\infty)$ and compute the integral." Here is my solution: $e^{-x}$ is continuous, hence measurable. We ...
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94 views

Weak derivative under the integral sign

Let $\Omega$ be a bounded and regular open subset $\Omega$ of $\mathbb{R}^N$ and $u:[0,\infty)\times \Omega\to \mathbb{R}$ be a smooth function (for example a smooth solution to a PDE). Thus the ...
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Determining if $f\in L^{p}(\mathbb R)$ from a bound on the measure of the level sets $\{|f|>\lambda\}$ for all $\lambda>0.$

$\textbf{The Problem:}$ Let $f$ be a measurable function on $\mathbb R$ with respect to the Lebesgue measure $m$. $\textbf{a)}$ Suppose that $$m(\{\vert f\vert>\lambda\})\leq(1+\lambda)^{-1}$...
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Lower bound on the integral of averaging operator.

In the following I denote by $B(x,r)$ the ball centered at $x$ of radius $r$, and by $|B(x,r)|$ the measure of this ball. I am trying to solve the following exercise. Question Let $f \geq 0, R \geq ...
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1answer
47 views

If $|f| < 1$, compute $\lim_{n\to \infty} \int \left ( \frac{f ^n}{1 + n |f|} \right )\, d\mu.$

Let $f: X \to \mathbb C$ be integrable in a measure space $(X, \mathfrak M, \mu)$, i.e. $\int |f| \, d\mu < \infty$. Suppose that $|f(x)| \leq 1$ for all $x \in X$. How can one compute the limit $$ ...
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61 views

Extensions of Riemann-Stieltjes without continuity problems

Are there any extensions of Riemann-Stieltjes integration that are able to handle the following integral? $\int_0^1 \alpha \space d\alpha$ where $ \alpha(x) = \left\{ \begin{array}{lr} 0 & ...
5
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1answer
161 views

limsup of continuous function is measurable.

I want to show that if $F$ is continuous on $[a,b]$ then $$\limsup_{h \rightarrow0, h>0} \frac{F(x+h)-F(x)}{h}$$ is measurable. By the definition of $\limsup$ we can write \begin{align} &\...
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136 views

Prove that $\lim_{n \to \infty} \int_{[0,1]}{x^n}\, dx = 0$. Where $\int$ represents Lebesgue integration.

Please check my proof, thank you. Prove \begin{align*} \lim_{n \to \infty} \int_{[0,1]}{x^n}\, dx = 0 \end{align*} Proof. Let $f_n(x) = x^n$. Since $f_n$ is a polynomial it is continuous and ...
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443 views

Weird use of Glasser's Master Theorem

Consider the following enumeration of the rational numbers in $[\,0,1)$: $$0, \frac{1}{2},\frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{3}{4}, \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}, \frac{...
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136 views

Are the canonical representatives of the Hilbert space $L^2$ basis-dependent?

The space $\mathcal{L}^p(\mathbb{R}^n)$ of functions $f$ such that $\int |f(x)|^p\, d^nx$ converges is only a seminormed rather than a normed vector space, because any function $f$ whose support has ...
5
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1answer
189 views

Integration by parts and non-absolutely continuous distributions

Let $x\in [a,b]$ be a real random variable with distribution $H$ that is not absolutely continuous (w.r.t Lebesgue measure). I saw this in a paper: $$ \int_a^b xH(dx) = b-\int_a^bH(x)dx. $$ I get it ...
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192 views

Are these different definitions of the upper integral really different?

I've come across at least three different definitions of the upper integral of a $[0,\infty]$-valued function with respect to an outer measure $\mu$. $\int_X^\ast f d\mu = \int_0^\infty \mu(\{f(x)>...
5
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1answer
287 views

Product of $L^{1}$ Function and Exponentially Integrable Function

Problem. Let $g\geq 0$ be in $L^{1}[0,1]$ and suppose that $\int gfdx<A$ whenever $\int e^{f}dx\leq 1$. What can one say about $|\{g>\lambda\}|$ for $\lambda\gg 1$. Is $g\in L^{2}[0,1]$...
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119 views

Show that if $f$ is measurable then $\{x\in A:c=f(x)\}$ is measurable for each $c$.

Let $A$ be a bounded measurable subset of $\mathbb{R}$. Show that if $f:A\rightarrow \mathbb{R}$ is measurable then $\{x\in A:c=f(x)\}$ is measurable for each $c$. Choose real $c$. Since $f$ is ...
5
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1answer
531 views

Absolutely continuous function on R

What is the definition of absolute continuity in whole $\mathbb{R}$. I know the definition on an interval $[a, b]$. I have a trouble with understanding the definition of absolute continuity in whole $...
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72 views

$L^2$ convergence of this sequence

I am given the following sequence of functions $(f_m)_{m \in \mathbb{N}}$. They are defined by $$ f_m(x):=\left( \frac{e^{-ix}-1}{-ix} \right)^m \left( \sum_{l \in \mathbb{Z}} \frac{\left|e^{-ix}-1 \...
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Convergence of the solution of Volterra integral equation with convergent kernel.

Consider the following Volterra integral equation $$ g(t) = \int_0^t K_n(t,s)w_n(s) ds $$ where $g(t)$ and $K_n(t,s)$ are known(continuous) and $K_n(t,s)\geq K_{n+1}(t,s)$ for all $t,s$. Moreover, $...
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A question about Lebesgue integral

Suppose f(x) is a Lebesgue integrable function on [0,1],and it has a least positive period 1. Define $S_k(x)$ by $$S_k(x)={1\over{2^k}}\sum_{i=1}^{2^k} f(x+{i\over{2^k}})$$ Try to show ...
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115 views

Prove that $F\in L^1(\mathbb{R})$

Let $f\in L^1(\mathbb{R})$ and continuous on $\mathbb{R}$ such that its Fourier transform $\hat f$ equals zero in a neighborhood of zero. Let $F$ be function such that $\hat F$ exists and $$\hat f(...
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30 views

If $f \in L^1[0,1]$ then there exists $c \in [0,1/2)$ s.t $\int_{c}^{c+1/2}f=1/2\int_{0}^{1}f(x)dx$

My attempt: Consider $$g(c)=\int_{c}^{c+1/2}f(x)dx$$g is continuous as $$g(c)=\int_{0}^{c+1/2}f(x)dx-\int_{0}^{c}f(x)dx$$ where both are continuous. Thus now we can use IVT to show such $c$ exists. ...
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In what sense is $\mathcal L^\infty$ the limit of $\mathcal L^p$ as $p\to\infty$?

In René Schilling's book Measures, Integrals and Martingales (second edition), a footnote on p. 116 says, Problem 13.21 shows that $\mathcal L^\infty$ is the limit of $\mathcal L^p$ as $p\to\infty$....
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Weird mistake that I cannot spot in a proof

I was doing this exercise: Suppose that $(X,\mathcal{S},\mu )$ is a measure space, $1<p<\infty $ and $f,g\in \mathcal{L}^p(\mu )$. Prove that Minkowski's s inequality is an equality if and ...
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87 views

Why dont we only consider complete measure spaces?

This question has been asked a few months ago, but in my opinion, it has not yet received a satisfying answer. So I ask again: Why should one consider non-complete measure spaces instead of requiring ...
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117 views

$\int_A f = \int_A g \implies f = g$ a.e.

Consider a measure space $(\Omega, \mathcal{F}, \mu)$ and let $f,g: \Omega \to \mathbb{R}$ be $\mathcal{F}$-measurable integrable functions. If $$\int_A f d \mu = \int_A g d \mu$$ for all $A \in \...
4
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49 views

Any Point in a Square is Covered by Finitely Many Disks

I'm working on the following problem. Let $A$ be a square of side length 1 in $\mathbb{R}^{2}$. Let $B(x_{k},r_{k})$ be disks centered at points $x_{k}\in A$ of radius $r_{k}$. Suppose that $\sum_{...
4
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0answers
175 views

Range of an integral operator

I tried to determine the range of the integral operator $T\in\mathcal{L}(L^2(0,1))$ given by $$(Tx)(t) = \int^1_t x(s)ds \text{ for } x \in L^2(0,1), t\in[0,1]$$ but it doesn't sound very logical. ...
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83 views

How do I use Fubini to show a set has the measure 0?

I am trying to show, that for fixed $v$ and $y \in \mathbb{R}^n$ the intersection of $L_y = \{y+tv ; t \in \mathbb{R}\}$ and a set E with n-dimensional measure 0, has 1-dimensional measure 0 for ...
4
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261 views

Line integral in terms of Lebesgue integral

Background I have been looking at the line integral and I am wondering why (or even if) it is well-defined in the context of Lebesgue integrals. The definition of wikipedia defines it like this (in ...
4
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202 views

Embedding of some function spaces

Consider the strictly monotone continuous function $d:\mathbb{R^+}\to\mathbb{R^+}$, and denote by $\mathcal{D}$ the space of all measurable functions $f:[0,1]\to\mathbb{R}$ such that: $$\int_0^1 d(|f(...
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135 views

Prove that $\int_Xf(x+y)dm(x) = \int_{X+y}f(x)dm(x)$

The following problem is from my exercise sheet. I was able to write a solution, but I am unsure in some passages, mainly in the sup equations in the second part, and I would be very grateful if ...
4
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72 views

$\mu(X)=\infty$, and $f\in L^p (X)$ for some $1<p<\infty \Longrightarrow f\in L^1 (X)$

What are some interesting examples of a measure space $(X,\mu)$ such that $\mu(X)=\infty$ and $f\in L^p (X)$ for some $1<p<\infty \Longrightarrow f\in L^1 (X)$ All the examples I have found ...
4
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140 views

Difficult Maximal Function Inequality

Let $1<p<\infty$. Suppose $a_i\geq 0$, and $\{B_{r_i}(x_i)\}$ is a sequence of open balls in $\mathbb{R}^n$ centered at $x_i$ with radius $r_i$. Let $g\in L^q(\mathbb{R}^n)$ where $\frac 1p+\...

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