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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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5 views

Anisotropic perimeter as derivative of measures.

Fix $K$ open bounded convex subset of $\mathbb R^n$ containing the origin and $\lvert x\rvert$ the euclidean norm of $x$. We define for every $x\in \mathbb R^n$ these quantities $$ \lVert x\rVert=\inf\...
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0answers
31 views

Lebesgue Differentiation Theorem for Continuous Functions

Let $g \in C_c(\mathbb{R})$. I want to prove the Lebesgue Differentiation Theorem does hold in this case, in particular: $\inf_\limits{\epsilon > 0}(\sup\{\frac{1}{m(B_r(x))}\int\limits_{B_r(x)}|\...
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1answer
31 views

How do I solve the volume bounded inside a solid [on hold]

Find the volume of the wedge in the first octant cut from the cylinder $$ y^2+z^2=4 $$by the $yz$ plane and the plane $y=x$. Indicated in the figure the slice used to compute the volume. I cant ...
2
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0answers
25 views

Find the norm of this operator

Consider the operator $A:\mathcal L_{2,w}(\mathbb R)\to\mathcal L_{2,w}(\mathbb R)$, which maps from the weighted $\mathcal L_2$-type space to itself. The operator acts in the following way: $$(Af)(t)=...
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1answer
28 views

Is the Lebesgue integral equal to the Lebesgue measure of the enclosed region?

Let $u:\mathbb{R}\to[0,+\infty]$ be Lebesgue integrable, and let $$A:=\{(x,y)\in\mathbb{R}^2 \; | \; 0\leqslant y\leqslant u(x)\}.$$ Thus $A$ is the set of all points enclosed between the graph of ...
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0answers
51 views

Let $f\in \mathscr{L}^1$. Show that for every $\epsilon$ there exists a continuous function $g$ such that $\int_X |f-g|d\mu < \epsilon$.

Let $f\in \mathscr{L}^1$. Show that for every $\epsilon$ there exists a continuous function $g$ such that $\int_X |f-g|d\mu < \epsilon$. Since the question is asking for a sequence of continuous ...
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0answers
42 views

Convergence in measure on a finite and infinite measure space

Let $(E,\mathcal{A},\mu)$ be a measure space. Let $(f_n)_{n \in \mathbb{N}}$ be a sequence of measurable functions, which converges in measure to $f:\forall \epsilon>0,\lim_n\mu(\left\{|f_n-f|>\...
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1answer
42 views

What is $\int_0 ^1 \int_x^1 \frac{f(t)}{t} dt dx $ if $\int f =1$?

Let $f$ be a Lebesgue integrable function on $[0,1]$ and $\int f = 1,$ and let $$g(x) = \int_x ^1 \frac{f(t)}{t} dt \quad x \in [0,1].$$ Calculate the integral of $g$. I feel like I'm supposed to ...
2
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1answer
28 views

Understanding the relation between the dominated convergence theorem and uniform convergence

Problem: Let $f \in L ^ { 1 } , | \widehat { f } | \in L ^ { 1 }$,$$ u ( x , t ) = \int _ { - \infty } ^ { + \infty } \widehat { f } ( \xi ) e ^ { 2 \pi i \xi x - 4 \pi ^ { 2 } a ^ { 2 } \xi ^ { 2 } t ...
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0answers
29 views

What is wrong with this product of measures in the $m\to \infty$ limit?

This question is about a passage in Reed & Simon concerning the Wiener measure and path integrals. To give the context, the authors consider the Trotter product formula $$e^{-t(H_0+V)}f=\lim_{m\...
2
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1answer
46 views

Grothendieck's lemma in $L^p$ spaces

So I am currently working on the proof of Grothendieck's Lemma : Let S $ \subset L^{\infty}(X) $, of finite measure, be a closed vector subspace of $L^p $ for a certain p such that $ S \subset L^{\...
4
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1answer
45 views

If $f_n \to f$ in $L^p$ and $g_n \xrightarrow{a.e.} g$ in $L^\infty$ then $f_ng_n \to fg$ in $L^p$

Exercise : Let $\Omega \subseteq \mathbb R^n$ be open and bounded, $\{f_n\}_{n \geq 1} \subseteq L^p(\Omega)$ with $1<p< \infty$ and $\{g_n\}_{n \geq 1} \subseteq L^\infty(\Omega)$. If it is $...
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0answers
68 views

Why can simple functions not take the value $\infty$?

When one develops the theory of integration, why is it usually the case that simple functions are not allowed to take the value $\infty$? Recall that if $(X,\mathcal{M})$ is a measureable space then ...
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1answer
24 views

Prove that the integration interchanging the order is the same

Let $X$ be a measurable space and $f:X\times[a,b]\to\mathbb R$ a function such that for all $t\in[a,b],f(\cdot,t)$ is measurable. If for all $x\in X, f(x,\cdot)$ is continuous over $[a,b]$ and if ...
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1answer
18 views

Some help with a measure-theoretic integration problem.

Let $g: [0, \infty) \rightarrow \mathbb{R}$ be continuous, increasing, and bounded with $g(0)= 0$ and $g(x) > 0$ for $x > 0$. Let $(X, \mathcal{A}, \mu)$ be a finite measure space and $f_{n}: X \...
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2answers
51 views

Proof $\frac{\sin x}{\sqrt{x}}$ is not Lebesgue integrable

I am trying to prove that the function $\frac{\sin}{\sqrt{x}}$ is not Lebesgue integrable on $[0, \infty]$. The proof I have seen seems to use a comparison test: \begin{align*} \int_0^{\infty} \left \...
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0answers
22 views

Is the dominated convergence theorem applicable whenever “THIS” theoem is applicable?

THIS theorem: Let $I =[a,b]$ be a closed and bounded interval and $\forall n\in \mathbb{N}$, $f_n:I \to \mathbb{R}$ be Riemann integrable on $I$. If the sequence $(f_n)$ converges uniformly to a ...
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0answers
56 views

Evaluate $\lim_{n \to \infty}\int_{0}^{1}\frac{n+\cos^n(e^x)}{4n+x^4} dx$

Evaluate $\lim_{n \to \infty}\int_{0}^{1}\frac{n+\cos^n(e^x)}{4n+x^4} dx$ Attempt: We define $f_n(x)=\frac{n+\cos^n(e^x)}{4n+x^4}$ on the domain $(0,1)$. This sequence of functions $(f_n)$ converges ...
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28 views

Motion of a particle defined via Lebesgue integral

Suppose $x_0 =0$. A particle moves as follows: $$x_t = \int_0^t a(s) ds$$ where $a: \mathbb{R}_+ \to \{-1,0,1\}$ is a measurable function. Suppose, I have that $a_s = 1$ if $x_s = 0$. I want to ...
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1answer
17 views

Reweighting preserves positive average

Let $h:[0,\infty) \to \Bbb R$ measurable and $\int_0^\infty \vert h(y) \vert \text d y <\infty$ and suppose $$\int_0^c h(y) \text d y \geq 0 \quad \forall c >0$$ Let $\omega : [0,\infty ) \to [0,...
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0answers
27 views

$\inf_{x \in E} f(x) = 0$ for $E$ a set of measure zero.

Let $(X, \mathcal{S}, \mu)$ be a measure space. I'm to explain why $\inf_{x\in E} f(x) = 0$ for every measure $0$ set $E$, where $\int f d\mu < \infty$ and $f \geq 0$. I know that $\int_{E} f d\...
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0answers
36 views

Let $(f_n)$ be a sequence in $M^+(\mathbb{R})$ proove that $ \displaystyle{\int \sum_{n=1}^{\infty}f_n = \sum_{n=1}^{\infty}\int f_n}$

How to prove the following exercise: Let $(f_n)$ be a sequence in $M^+(\mathbb{R})$ proove that $ \displaystyle{\int \sum_{n=1}^{\infty}f_n = \sum_{n=1}^{\infty}\int f_n}$ it reminds me the ...
-1
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1answer
41 views

Give an example of strict inequality in Fatou's Lemma

The fatou's Lemma says Let $ \left\{ f_n, n = 1,2,...\right\} $ be a sequence of non-negative measurable functions. Then $$ \liminf \int f_n \geq \int \liminf f_n$$ Some hint? I think this Showing ...
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2answers
44 views

Prove that $f$ is not Lebesgue integrable

I need a hand with the following exercise: Prove that $f: (0,2) \to \mathbb{R}$ given by $f(x) = \begin{cases} \frac{1}{x} & 0<x\leq 1 \\ \frac{1}{x-2} & 1< x < 2 \...
3
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1answer
34 views

A limit of an integral of a quotient related to fractional Sobolev space

Let $0<\alpha<1$ and $1\leq p<\infty$. Suppose $f\in L^p(\mathbb{R}^n)$ satisfies \begin{align} \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{|f(x)-f(y)|^p}{|x-y|^{n+\alpha p}}dxdy<\infty \...
2
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2answers
44 views

Sequence of measures in L-p Spaces

Let $(X,\mathscr{A},\mu)$ be a measure space and $\{E_n\}_{n \in \mathbb{N}} \subset \mathscr{A}$ such that $$X=\bigcup_{n=1}^{\infty}E_n \quad \text{and} \quad E_n \subset E_{n+1} \quad(\forall n\in ...
1
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1answer
32 views

Convergence of integral with compact support

Let $a,b \in \mathbb{R}$ with $a<b$ and let $f:[a,b] \longrightarrow \mathbb{R}$ be continuous. Show that $$\int_a^bf(x)\text{cos}(nx)\:dx \longrightarrow 0 \quad as \quad n \longrightarrow \infty.$...
3
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1answer
83 views

Show that $\int_\mathbb{R^n} F(|g(x)|) dx = - \int_0^{\infty} F(\alpha) d \lambda(\alpha) = \int_0^{\infty} F'(\alpha) \lambda(\alpha) d\alpha$

So, I've been tasked with proving the following: Show that if $F$ is a non-negative differentiable function defined on $[0, \infty)$ with $F(0) = 0$ and $g$ is a measurable function on $\mathbb{R^n}...
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0answers
28 views

if and only if condition for a function to be a Lebesgue Integrable.

Given a function $f(x)$ over any closed interval $[a,b]$, does there is any if and only if condition so that function will be Lebesgue Integrable. For Riemann integrability there is a condition that ...
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214 views
+200

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 ...
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1answer
67 views

How do we need to apply the martingale convergence theorem here?

Let $(E,\mathcal E,\mu)$ be a measure space $E_0\in\mathcal E$ with $\mu(E_0)\in(0,\infty)$ $n\in\mathbb N$ $B_1,\ldots,B_n\subseteq\left.\mathcal E\right|_{E_0}:=\left\{B\cap E_0:B\in\mathcal E\...
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0answers
36 views

Showing the equality of two integrals

If $B$ is a finite ball in $\mathbb{R^n}$ and $f \in L^1(B)$, why do we have the following equality: $$\int_1^\infty \frac{1}{t}(\int_{\{|f| > \frac{t}{2}\}}|f(x)|dx) dt = \int_B ( \int_1^{\text{...
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1answer
15 views

Rewriting expectation (Lebesgue)

I have the following: Let $F(x)$ be the cumulative distribution function of the random variable $x$ for which it holds that $Pr(x>0)=1$, i.e. $F(a)=\int_{-\infty}^{a}f(x)dx$. As a result, I have:...
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1answer
23 views

Locally integrable function and limits

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a locally Lebesgue integrable function, ie $f$ is Lebesgue measurable and $$\int_{K}|f(x)|dx<+\infty$$ for every compact subset $K\subset\mathbb{R}^n$. Is it ...
1
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1answer
63 views

Lebesgue Integral equal to Lebesgue Integral over partitions

I did the proof though I didn't use the hypothesis of $X=E\cup F$. Also I didn't see the why of $f$ to be in $\overline{\mathbb R}.$ Could this be any $Y$ space? Let $f:X\to\overline{\mathbb R}$ be ...
2
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1answer
29 views

Showing that the bounded $C \subseteq L^p[0,1]$ is uniformly integrable.

Exrcise : Let $C \subseteq L^p[0,1], 1 < p < \infty$ be bounded. Show that $C$ is uniformly integrable. Attempt : It is $L^p[0,1] \subseteq L^1[0,1]$ and $L^p[0,1] \hookrightarrow L^1[0,1] \...
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0answers
48 views

Lebesgue integration of a sine function

I'm trying to understand this problem but it does not go very well. I don't understand how I should calculate this integral, when I'm given the density function. Read the problem below: "Let $\lambda$...
0
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1answer
11 views

Pointwise convergence to a function with infinite integral implies the integral is eventually negative?

Suppose $f_n$ converges pointwise to $f$ as $n\rightarrow\infty$, and that $\int{f}=\infty$. Must it be the case that there is some $N$ such that for $n>N$, $\int{f_n}>0$? Or at least that $\...
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0answers
26 views

Inequality for two Hardy-Littlewood Maximal Type Operators

$\textbf{The Problem:}$ Let $$Mf(x)=\sup\limits_{x\in B}\frac{1}{m(B)}\int_{B}\vert f\vert\quad\text{for }x\in\mathbb R^d$$ denote the Hardy-Littlewood maximal function, where the supremum is ...
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2answers
37 views

$L^1$-Cauchy sequence of step functions that converges a.e. also converges in $L^1$

Let $X$ be a measure space and $E$ be a Banach space. A step function from $X$ to $E$ is a measurable function ($E$ takes the Borel measure) with finite image and whose support has finite measure. ...
0
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1answer
36 views

Is every a.e. limit of step functions also an a.e. limit of a Cauchy sequence of step functions?

Let $X$ be a measure space and $E$ be a Banach space. A step function from $X$ to $E$ is a measurable function ($E$ takes the Borel measure) with finite image and whose support has finite measure. ...
0
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1answer
12 views

Why does Lang need finite measure support in the proof Lemma 3.2 (Real and Functional Analysis)?

If anyone has the book, Lemma 3.2 and its proof are on pages 130-132. Here is the statement of the lemma: Let $\{g_n\}$ and $\{h_n\}$ be Cauchy sequences of step mappings of $X$ into $E$, ...
0
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1answer
28 views

Is the integral of the product of two real-valued functions an inner product for every kind of integral?

I'm wondering if the integral of the product of two real-valued functions on a given space is also necessarily also an inner product for every function that can be defined on that space. E.g., are the ...
4
votes
1answer
87 views

Interesting Difference between Lebesgue and Riemann Integral

The Riemann integral makes it so that if we have $|f| \leq |g|$ on $[0,1]$, then integrability of $g$ does not necessarily imply the integrability of $f$. For example, let $f = \chi_\mathbb{Q}$, $g = ...
2
votes
1answer
61 views

If $f\in L^{1}([0,1])$ and $F(x)=\int_{0}^{x}f$ for $x\in[0,1],$ prove $F$ is absolutely continuous from the definition

$\textbf{The Problem:}$ Let $f\in L^{1}([0,1])$ and define $$F(x)=\int_{0}^{x}f\quad(x\in[0,1]).$$ Prove directly from the definition of absolute continuity that $F$ is absolutely continuous. $\...
4
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0answers
39 views

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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0answers
18 views

Using the Lebesgue Differentiation Theorem

Let $f\in L^1(\mathbb{R})$ and let $g(x) = \int_{-x}^{x} f(y) dy$. I want to show that g is differentiable almost everywhere and compute $g'$. One way I've done this is showing that $g$ is absolutely ...
2
votes
1answer
59 views

Proving that given a measure, the function $f \in L^{1}$

I got this problem I cant figure out by the definitions Im given :/. Any hint will be really appreciated! Let $\Omega=\{1,2,...\}$, $F=2^{\Omega}$ and $\mu$ defined over $(\Omega, F)$ by $\mu(k)=p(1-...
0
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1answer
32 views

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

Let $f\in L^{1}(\mathbb{R})$ and $\forall x\in\mathbb{R},$ let $F(x)=\displaystyle\int_x^{x+1} f(t)\ dt$. Prove that $F\in L^{1}(\mathbb{R})$. Hint: Use Tonelli. Attempt: I indeed used Tonelli's ...
4
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1answer
79 views

Does $\displaystyle\int_\mathbb{R} f_n\ dm\to \displaystyle\int_\mathbb{R} f\ dm$?

Let $f_n$ be a sequence of measurable functions on $\mathbb{R}$ converging a.e. to $f$. If $0\leq f_n\leq f$ a.e. Does it follow that $\displaystyle\int_\mathbb{R} f_n\ dm\to\displaystyle\int_\mathbb{...