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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Rewriting Lebesgue measures and Riemann integrals

Question Let $\mu$ be a Lebesgue measure on $(\mathbb{R}, \mathcal{B})$. Show that for any continuous function $f: \mathbb{R} \to \mathbb{R}$ and any $-\infty<a<b<\infty$, the following holds....
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How to transform using the term-by-term integration theorem?

Question The measure $\mu$ is defined on the measurable space $(\mathbb{N},2^\mathbb{N})$ as follows. $$ \mu(A) := \sum_{k\in A} \frac{1}{k} $$ The following equality follows when term-by-term ...
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1 vote
1 answer
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$F$ is Lipschitz $\Longrightarrow$ $F(x)=F(a)+\int_{a}^{x}f(t)\,dt$

According to the Fundamental Theorem of Lebesgue Integral Calculus, the following statement holds: Let $F:[a,b]\to\mathbb{R}$. Then $F$ is absolutely continuous iff there exists a function $f:[a,b]\...
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1 vote
1 answer
60 views

A Lebesgue Integrable Function

let $f \geq 0$ be a finite $a.e.$ on $E$ with finite measure. Let $\epsilon > 0$ and consider the following partition of $\mathbb{R}_{\geq 0}$ for $n\geq 0$ $$0 = x_0 < x_1 < ... < x_n <...
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1 answer
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How to Calculate the Lebesgue Integral?

Question $\psi(x, y) = \psi_x(y) = e^{xy}, \cosh{x} = \frac{e^x + e^{-x}}{2}$. Why is the following valid? What is the detailed calculation? $$ \cosh{x} = \int_\mathbb{R}\psi_x d\mu,\ \mu := \frac{1}{...
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4 votes
2 answers
60 views

How to handle measurements and Lebesgue integrals? (with concrete example)

Question Define the measure $\mu$ on the measurable space $(\mathbb{R}, \mathcal{B})$ as follows. $$ \mu\left( (a, b] \right) = \int_a^b \frac{1}{1+x^2}dx\ \ \mathrm{for\ all\ interval\ (a,b]}. $$ ...
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35 views

How to verify that $\chi_{\{x \in X; f(x) > t \}}$ is integrable?

Question Let $(X, m, \mu)$ be a measure space and $f: X \to \mathbb{R}$ a nonnegative measurable simple function. I want to show that $\chi_{ \{ x \in X; f(x) > t \} }\ (t > 0)$ is integrable at ...
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Integral and inequality

Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$ Let $\mathcal{E}:=\{\phi \in C_c^{\infty}(\mathbb{R}),\text{supp}(\phi) \subset B(0,1),||\phi||_{\infty} \leq 1\}.$ Prove ...
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1 answer
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What makes a continuous function on open interval integrable?

Let $a < b$. Let $f: (a, b) \rightarrow \mathbb{R}$ be continuous. What is a sufficient condition to make $f$ integrable (in the Lebesgue sense) on $(0, 1)$? For example, $f: (0, 1) \rightarrow \...
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What is the correct way of translating a joint probability to Lebesgue-Stieltjes integrals

I have a set inequalities of several non-negative independent random variables and I want to compute the probabilities from these. While the cumulative distribution function of each of these random ...
1 vote
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Fundamental lemma for non-negative functions

is there some sort of fundamental lemma of calculus of variations for non-negative functions? Let $u\geq 0$ and $u\in L^1(\Omega)$ on a domain $\Omega$ in $\mathbb{R}^N$. If $$\displaystyle\int_\Omega ...
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Definition of Lebesgue integral in Reed and Simon.

In the book "methods of modern mathematical physics I functional analysis" by Reed and Simon, the Lebesgue integral is defined as \begin{align} \int f(x)dx=\lim_{n\to\infty} \sum_{2^n}(f) \...
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1 answer
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Absolutely convergent series of Lebesgue integrals of $f$ on intervals summing to $[1,\infty)$ implies $f$ is lebesgue integrable on $[1,\infty)$. [closed]

Prove or find counterexample. If series $\sum_{n=1}^{\infty} \int_{n}^{n+1}fdm$ is absolutely convergent then $\int_{1}^{\infty}fdm$ is lebesgue integrable, meaning one of $\int_{1}^{\infty}f^+dm$, $\...
0 votes
0 answers
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Divergence theorem for Sobolev functions on whole space

Let $u_1, \dots, u_d, w \in H^1(\mathbb{R}^d).$ We set $u:=(u_1, \dots, u_d)^{\top}.$ Is the following identity true for the whole space? (If we assume the divergence theorem for Sobolev functions on ...
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4 votes
1 answer
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Question in the proof of Lusin's theorem.

I want to prove Lusin's theorem. Let $E\subset\mathbb R$ be a measurable set and $f:E\to\mathbb R$ be a measurable function. Then, for each $\epsilon$, there exists $F\subset E$ s.t. $F$ is closed ...
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1 vote
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Monte Carlo integral over infinite discrete set

In many articles (e.g. in Wikipedia https://en.wikipedia.org/wiki/Monte_Carlo_integration ), Monte Carlo integration is introduced as an numerical approximation of an integration over $n$ dimensional ...
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Is there Dominated Convergence Theorem for Riemann integral?

I learned there is Bounded Convergence Theorem for Riemann integral: (Arzela’s theorem) If $(f_n)_{n=1}^\infty$ is a sequence of Riemann integrable functions that are bounded by the same constant $M&...
2 votes
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Prove there exists measurable $B_k$ s.t. $\int f d\mu = \sum_{k=1}^n \frac 1 k \mu(B_k)$

Let $\mu$ be a (Borel?) measure on $\mathbb R$ and $f: \mathbb R \to \mathbb R^+$ be $\mu$-measurable. An exercise asks me to prove that there exists measurable finite family $(B_k)_{k=1,\dots,n}$ ...
0 votes
2 answers
54 views

Integration vs sum over dense subsets

Why is the Lebesgue integral can't be written as an infinite sum of the form (or can it ?) $$\int_0^1 f(t)dt= \sum\limits_{i=1}^\infty {\lambda(f)}_i f(t_i)\quad , \ \forall f\in C([0,1])$$ where the $...
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1 answer
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Proof that neither “almost none” nor “almost all” functions which are Lebesgue measurable are non-integrable

According to answer in (https://mathoverflow.net/a/28114/87856) and (https://www.ams.org/journals/bull/2005-42-03/S0273-0979-05-01060-8/S0273-0979-05-01060-8.pdf) "almost all" can be defined ...
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1 vote
1 answer
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Making sense of dirac delta of a function $\delta(\psi) = \int \mathscr{D}\varphi e^{-i\int\bar{\varphi}\psi}$

I am currently trying to learn about the MSR Formalism, which rephrases SDE problems in terms of path integrals. This is in the context of turbulence for plasma physics; I know that path integrals do ...
0 votes
1 answer
35 views

How to understand if a function is in $L^p$ space? [closed]

I'm studying measure theory and I'm having some trouble understanding which functions are in $L^p$ space. For example the constant function $f(x) = c$ is in $L^1$ space? Following the definition I ...
1 vote
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If $X$ is a vector distributed according to $e^{-\frac ET}$, what's the distribution of the components $X_i$?

Let $\lambda$ denote the Lebesgue measure on $\mathcal B(\mathbb R)$, $d\in\mathbb N$, $D:=[0,1)^d$, $p:D\to(0,1]$ be a Lebesuge integrable function with integral $c_p>0$, $\sigma:=p^{-\frac1d}$, $...
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1 vote
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28 views

Calculating Lebesgue-Integral of Function over Annulus

I came across this exercise in 'Analysis $3$' by Otto Forster. I will have measure and integration theory in my 3rd semester so right now, I am not really informed about the preliminaries, to solve ...
3 votes
0 answers
30 views

Construct a sequence of Lebesgue integrable functions.

Im studying for my qualifying exam and I encounter this old problem: Construct a sequence of integrable functions $f_n : [0, 1]\to\mathbb{R}$ such that a. $f_n$ converge to zero function on [0, 1] ...
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Calculate $\int_{[0,1]}xdm$ by Lebesgue integral definition of positive functions

Calculate $\int_{[0,1]}xdm$ by Lebesgue integral definition of positive functions Consider the sequence of simple functions $$\phi_n=\sum_{k=0}^n\frac{1}{2^{k+1}}\mathcal{X}_{[\frac{1}{2^{k+1}},\frac{...
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1 vote
2 answers
50 views

Integration over a Translated Set is Uniformly Continuous?

I'm looking for help with the following problem (working with Lebesgue measure here): Let $ψ∈L^\infty(R)$ and let $B \subseteq \mathbb R $ be measurable such that $m(B) < \infty$. If $B+x =\{b+x|b∈...
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0 votes
1 answer
58 views

Sufficient assumptions to obtain boundedness of integral

we have $$\displaystyle\int_{E\times(0,T)}f(x,\tau)^{p-1}\partial_\tau g(x,\tau)\,dx\,d\tau$$ with $f\in C(0,T;L^p(E))$ for some domain $E$ in $\mathbb{R}^n$, $p>1$ and $g\in W^{1,p}(0,T;L^p(E))$. ...
9 votes
2 answers
144 views

A non-zero function on $\mathbb{R}^n$ whose integral over any ball of radius $1$ is zero?

Does there exist a measurable function $f:\mathbb{R}^n\rightarrow\mathbb{R}$, for some $n>1$, which is not equal to $0$ almost everywhere such that for any ball $B$ of radius exactly $1$ we have $$\...
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0 votes
0 answers
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Convergence of integrals for subsequence

In an article I'm currently reading, a reasoning is used that I don't understand. We have an integral of a function over a domain with both depending on the same $\epsilon>0$. They show that $$\...
0 votes
0 answers
29 views

Evaluation of Sobolev ($L^p$) function actually well defined?

I will ask a similar question to the one from yesterday. If we have $$\displaystyle\int_Ef(x,t)|^{\tau_2}_{\tau_1}\,dx$$ where $f$ is a Sobolev function (or maybe even just some $L^p$ function) (not ...
3 votes
0 answers
21 views

Proving a basic form of monotone convergence

Problem: Definition 4.4 of Probability Theory: A Comprehensive Course by Klenke says that If $f \colon \varOmega \to [0, \infty]$ is measurable, then we define the integral of $f$ with respect to $\...
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Choosing alway simple functions on rectangles which converge against some function [closed]

Let $(X,\Sigma,\mu)$ and $(Y,\Gamma,\nu)$ be propability spaces. Consider $f\in L_{1}(X\times Y,\Sigma\otimes\Gamma,\mu\otimes\nu)$. Is it always possible to find a decreasing sequence say $f_{N}(x,y):...
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Integral not defined in 0

My teacher proposed us the following challenge: Provide conditions for the reals numbers $a$ and $b$ such that the following integral converges and compute the value: $$ \int_0^c \left|\frac{1}{t^a}\...
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6 votes
1 answer
152 views

Is Lebesgue integral a continuous functional in this context?

Statement: I asked a similar question before which was closed. I tried to provide some context in this version. Let $p$ be a probability measure whose support is $\left[a,b\right]$. Consider the set ...
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Is $x \mapsto \mu( \text{supp}f_x)$ continuous?

Let $X$ be a measurable space equipped with a complex Borel measure $\mu$, and let $\{f_x\}_{x\in X}$ be a family of continuous functions $f_x:X\rightarrow \mathbb{C}$ such that the function $X\...
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1 vote
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Lebesgue integral domain help [closed]

$f(x)=\left\{ \begin{align} & \frac{1}{{{x}^{2}}},\,\,\,\,\,\,x\ne 0 \\ & \infty \,\,\,\,\,\,\,\,\,x=0 \\ \end{align} \right\}$ I want to show that this function is Lebesgue Integrable. ...
0 votes
2 answers
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$g(x,t) =f(x+t)$ is Lebesgue measurable if $f$ is Borel measurable

Let $f: \Bbb R \to \Bbb R$ be Borel-measurable. $(\mathrm{A})$ Show that $g:\Bbb R ^2 \to \Bbb R, g(x,t)=f(x+t) $ is Lebesgue-measurable. $(\mathrm{B})$ Assume $g \ne 0$ (meaning $g$ is not a.e. zero)....
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separable measure algebra

We now that any separable measure algebra $A_{\mu}$ with total mass equal to one, without any atoms, is isomorphic to the unit intervall with the lebesgue measure. This is a classical result of ...
5 votes
1 answer
102 views

Show that $\displaystyle \lim_{k\rightarrow \infty}f(k^\alpha x) \rightarrow 0$

Let $f$ be an integrable function over the positive real numbers. Prove that $f(k^\alpha x) \rightarrow 0$ whenever $k \rightarrow \infty$ for almost everywhere $x>0$ and any real number $\alpha>...
0 votes
0 answers
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Subspace of Banach have Lebesgue property

Let Y be a subspace of X. If X has the property of Lebesgue, then Y has the property of Lebesgue. Firstly, let us discuss about property of Lebesgue: A Banach space X has the property of Lebesgue if ...
1 vote
0 answers
28 views

speedup computation of integrals on a grid

Let $d\in\mathbb N$, $D\in\mathcal B(\mathbb R^d)$ be bounded, $p:D\to[0,\infty)$ be Lebesgue integrable, $\sigma>0$, $$\varphi(x,y):=p(x)e^{-\frac{\|x-y\|^2}{\sigma^2}}\;\;\;\text{for }x,y\in D$$ ...
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0 votes
0 answers
38 views

A simple proof of Lebesgue differentiation theorem for dimension $1$

Let $\lambda$ be the Lebesgue measure on $[a, b]$ and $f:[a, b] \to \mathbb{R}$ $\lambda$-integrable. We define $F:[a, b] \to \mathbb R$ by $F(x) := \int_a^x f \mathrm d \lambda$ for all $x \in [a, b]$...
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1 vote
0 answers
30 views

Lebesgue integration of fractional part in on $L^1[0,1]$

I was trying to solve the following problem: Suppose $h\colon [0,1]\to \mathbb{R}$ is a continuous function such that $\int_0^1 h(x)dx=0$. For $x\ge 0$, let $\{ x \}$ represent the fractional part of $...
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3 votes
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$L^2$ integrability of a function of two variables

I have to study the $L^2(\Omega)$ integrability ($\Omega\subset\mathbb{R}^2$ bounded) of the function: $$ f(x,y)= \begin{cases} \dfrac{1}{(1+y^2)|x|^{1/3}} & x\neq 0\\ \\ 0 & x=0 \end{cases} $$...
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1 answer
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Why is this parametric integral not continuous?

Define $f(x, w) = x \exp(-(xw)²)$ and $g(x) = \int_\mathbb{R} f(x,w) d\lambda(w)$. It is easy to check that for $x > 0$ we have $g(x) = \sqrt{\pi}$ by using the substitution $t(w) = xw$ and ...
0 votes
0 answers
37 views

If $\int_a^x f(t) \mathrm d t=0$ for $\lambda$-a.e. $x \in[a, b]$, then $f=0$ $\lambda$-a.e.

Let $\lambda$ be the Lebesgue measure on $[a, b]$. We endow $[a, b]$ with the Lebesgue $\sigma$-algebra $\mathcal L([a, b])$. Previously, I proved that Theorem Let $f:[a, b] \rightarrow \mathbb{R}$ ...
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3 votes
1 answer
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Durrett Exercise 1.4.2

1.4.2 Let $f\geq 0$ and $E_{n,m}=\{x: m/2^n \leq f(x) \leq (m+1) /2^n\}$. As $n \uparrow \infty$, show that: $$\sum_{m=1}^{\infty} \frac{m}{2^n} \mu(E_{m,n}) \uparrow \int f d\mu$$ I was trying to ...
2 votes
0 answers
59 views

If $\int_a^x f(t) \mathrm d t=0$ for all $x \in[a, b]$, then $f=0$ $\lambda$-a.e.

Let $\lambda$ be the Lebesgue measure on $[a, b]$. We endow $[a, b]$ with the Lebesgue $\sigma$-algebra $\mathcal L([a, b])$. Then $([a, b], \mathcal L([a, b]), \lambda)$ is a complete measure space. ...
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2 votes
1 answer
34 views

Extract a uniformly integrable sequence

Let $(X,\mathcal A,\mu)$ be a measurable space. A sequence $\{f_n\}$ of positive measurable functions is said to be uniformly integrable if for any $\epsilon$, there exists a $\delta$ such that if $U$ ...
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