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.
7,931 questions
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Generalizing integration of $[0,+\infty]$ valued functions
Several of the most basic results of Lebesgue integration are stated for nonnegative (Lebesgue measurable) maps $X \to [0,+\infty]$ and may involve the order structure $\leq$ of $[0,+\infty]$:
Fatou'...
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When is it justified to take a limit inside a series? [closed]
Let
$$
f(x)=\sum_{k=1}^{\infty}(-1)^k(k+1)\,\chi_{\left(\frac1{k+1},\,\frac1k\right]}(x),
\qquad x\in(0,1].
$$
Thus $f$ is constant on each interval $\left(\frac{1}{k+1},\frac{1}{k}\right]$, taking ...
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When does $\int_0^{1/2} f(x) \ dx= \int_0^1 f(\varphi(x))\,v(x) \ dx$ hold for $\varphi(t)=\min\{t,1-t\}$?
Let $\varphi:[0,1] \longrightarrow [0,\frac{1}{2}]$ with $$\varphi(t)= \min\{t, 1-t\}$$
I need to prove, $$\int_0 ^{1/2}f(y) \ dy =
\int_0^1 f(\varphi(x))v(x) \ dx$$ if and only if $v \in \mathcal{L}...
1
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1
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$\int f(x \alpha) dx =\left| \alpha \right|^{-1} \int f(x) dx$ $f$ a measurable positive or integrable function
I came across this question in the Book Introduction to Measure Theory by Irribarren. Specifically, on its chapter about integral with respect to any measure.
$f$ a positive measurable function or ...
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Proof that the convolution is finite $\mu$-a.e
With respect to exercise 10.R of Bartle's integration elements, it is requested to see that the convolution is certainly finite, it is detailed as follows:
Let $f$ and $g$ be Lebesgue integrable ...
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Why are we satisfied to explain the power of Lebesgue integration just by saying it is ''horizontal, rather than vertical''?
A standard intuition found in textbooks for the power of the Lebesgue integral compared to its Riemann counterpart is that "We integrate by taking horizontal slices, rather than vertical ones.&...
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measurability definitions in time-dependent Lebesgue spaces
I have some questions regarding the notion of measurability for functions $f : (0,T) \to L^q(\mathbb{R})$ that belong to spaces like $L^p(0,T; L^q(\mathbb{R}))$.
How is measurability of such a ...
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How do I prove the change of variables into polar coordinates using measure theory?
From this answer I have that $ \int_Yf(y)\,\mathrm{d}(g\mu)(y)=\int_Xf(g(x))\,\mathrm{d}\mu(x)$, where $g$ is a map between measurable spaces and $g\mu$ is the image measure.
With $X=[0,r]\times[0,2\...
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Existence of constants on bound of $L^1,L^p$ function.
Let $f\in L^1(\Bbb{R})$ and suppose $f \in L^p(\Bbb{R})$ for some $p>1$. Show there exists some $c>0$ and $\alpha \in (0,1)$ such that
$$\int_A \vert f \vert \leq cm(A)^\alpha$$
for all $A \...
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Algebraic integral [closed]
I noticed that integration by substitution is essentially the ``sprinkler rule'' (distributivity of a composition over an action).
I wrote an article where:
The integral is treated as a monoid ...
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An application of MCT when given a decreasing sequence. [duplicate]
Let $\{f_n\}\subset L^+(\Bbb{R}^n)$ be decreasing with $\lim_{n \to \infty}f_n=f.$ and if $\int f_1<\infty$, then $\lim\int f_n = \int f$.
Attempt:
My thoughts were MCT but I have decreasing.
So ...
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Integrating over a plane when variables satisfy $y = f(x)$: does $\displaystyle \iint_{\mathbb{R}^2} F(x,y)\,u(x,y)\,dx\,dy$ make sense?
Let $u:\mathbb{R}^2 \to \mathbb{R}$ be a bounded function with compact support, and let
$F:\mathbb{R}^2 \to \mathbb{R}$ be measurable.
Consider the double integral
$$
\iint_{\mathbb{R}^2} F(x,y)\,u(x,...
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Dominated convergence theorem: almost everywhere condition
Let $f_n$ be a sequence of integrable functions with $|f_n(x)| \leq g(x)$ for some integrable function $g$. If $f_n$ converges to a measurable function $f$ almost everywhere, then $f$ is integrable ...
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For any non-negative measurable function, is there an arbitrarily small set where its integral is arbitrarily small?
The book "Integración de Funciones de Varias Variables" by José Antonio Facenda and Francisco José Freniche is in Spanish. The following exercise appears in Chapter 2:
Let $f: \mathbb{R^n} \...
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Monotone convergence theorem proof
I'm trying to follow Bartle's proof of the Monotone Convergence Theorem in measure theory.
If $f_n$ is an increasing sequence of measurable functions converging to $f$, then
$$\lim \int f_n \,d\mu = \...
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showing $\lim_{a \to \infty} a\int_0^1 e^{-ax}f(x)dx=f(0)$ [duplicate]
Show for all $f \in C([0,1])$ that
$$\lim_{a \to \infty}a\int_0^1 e^{-ax}f(x)dx=f(0).$$
My Thoughts:
First I realized, $m([0,1])=1<\infty$ and if we put $g(x)=e^{-ax}f(x)$, then $g(0)=f(0)$ and $g(...
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Integrating a function of x with respect to y
If I want to integrate a function $f(x)$ with respect to $y$:
$$ \int f(x) \, dy $$
where $y = f(x)$, does $x$ act as a constant? Meaning will the constant rule of integration
$$ \int k \,dx = kx + C$...
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Measure Theoretic Definition of the Integral
I am currently reading through Analysis by Lieb and Loss. They define the integral of a function in the following way:
Suppose that $f : \Omega \to \mathbb{R^+}$ is a nonnegative real-valued ...
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If $\left\vert\int fg\right\vert\leq\left(\int|g|^p\right)^{\frac{1}{p}}$ for some $p\in(0,1)$, then $f=0$ a.e.
Let $f\in L_{\text{loc.}}^1(\Bbb{R}^d)$ such that for some $p\in (0,1)$,
$$\left\vert\int f(x)\,g(x)\,\mathrm dx\right\vert\leq\left(\int\vert g(x)\vert^p\,\mathrm dx\right)^{\frac{1}{p}}$$
for all $g\...
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Is this exchange of summation and integration implication true? [closed]
Let $ f_k \in L^2(\mathbb{R})$ for every $ k \in \mathbb{N} $, and suppose that
$$\sum_{k \in \mathbb{N}} \|f_k\|_2 < \infty.$$
Does this imply that:
$$
\sum_{k \in \mathbb{N}} \int_{\mathbb{R}} ...
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Find $ \lim_{n \rightarrow +\infty} \left(n(-1)^n \sum_{k=1}^n \frac{(-1)^k}{k (n-k)!}\right)$
I want to calculate the value of
$$ \lim_{n \rightarrow +\infty} \left(n(-1)^n \sum_{k=1}^n \frac{(-1)^k}{k (n-k)!}\right)$$
which is a suggested problem in Folland's Real Analysis book. It is ...
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An Exercise in Condition Distribution that does not Go Right
Consider the following exercise. Let $X$ be a random variable such that, knowing $\sigma > 0$, it has a conditional distribution of $N(0, \sigma^2)$. We give $\sigma$ a distribution of Lebesgue ...
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“Finite” version of convergence theorems
One usually sees the fundamental convergence results of measure theory stated for function with values in the extended real line $[-\infty,+\infty]$. This entails spending some effort extending the ...
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A question about the integral of Marcinkiewicz
Here, we are interested in the following integral ; where $F$ is a fixed closed subset of $\mathbb{R}^d$.
$$ I(x)=\int_{B(0;1)}\frac{\text{dist}(x+y,F)}{|y|^{d+1}}dy ,~ x\in\mathbb{R}^d. $$
The above ...
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$\int f\,d(\mu+\nu)=\int f\,d\mu+\int f\,d\nu$ for complex measures?
Let $X$ be a measurable space. Let $\mu$ and $\nu$ be complex measures on $X$. Let $f\in L^1(\mu)\cap L^1(\nu)$. Then is it true that $f\in L^1(\mu+\nu)$ and $\int f\,d(\mu+\nu)=\int f\,d\mu+\int f\,d\...
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Is a real-valued convex function measurable on a closed interval an integrable on an open interval?
Let $f:[a, b]\rightarrow\mathbb{R}$ be convex. Then it is continuous (on $(a,b)$) but it may not be continuous on $[a,b]$. If I understand correctly, it will be measurable on $(a, b)$ (following from ...
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Show an $L^1$ function is a.e. $0$ avoiding using Lebesgue Differential Theorem
This is a test-problem on real analysis. Let $f\in L^1(\mathbb R)$ satisfy $$\limsup_{\epsilon\to 0+} \int_{\mathbb R} \int_{\mathbb R} \frac{|f(x)||f(y)|}{|x-y|^2+\epsilon^2} dxdy<+\infty.$$
Show ...
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Lebesgue differentiation theorem and l'Hôpitals rule
Let $f, g$ be $L^1$ functions. Define $I_g(x) = \int_x^\infty g(t) \, \mathrm d t$ where $\mathrm d t$ is the Lebesgue measure, and similarly for $f$. It is not hard to show that $$ \frac{\mathrm d }{\...
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Modulus of Lebesgue Integral of Function and Integral of Modulus of Function (Papa Rudin RCA Theorem 1.33)
Please see below the main statement from Papa Rudin RCA Theorem 1.33 (The full theorem 1.33 and proof are given in the linked question below):
If $f \in L^1(\mu)$, then $$\left|\int_X f \,\mathrm d\...
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Does Fubini-Tonelli hold as long as the Lebesgue integral is defined?
Background
Fubini’s and Tonelli’s theorems describe when we can interchange the order of integration in a double integral. Specifically:
Fubini's theorem states that if a measurable function $ f(x, y)...
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Inequality to bound integrals of the form $\int_{Z}(y-f(x))^p d\rho$.
Consider a probability measure $\rho(x,y)$ on $Z=X\times[-M,M]$, where $M>0$ and $X$ is a compact subset of $\mathbb{R}$.
We assume that $\rho(x,y)=\rho(y|x)\rho_X(x)$, where $\rho(y|x)$ is the ...
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Doubt in an equality between two integrals
My probability professor wrote the following equality in class: $\int_0^{+\infty}(\int_0^yg(\lambda)d\lambda)\mu_X(dy) = \int_0^{+\infty}(\int_{\lambda}^{+\infty}\mu_X(dy))g(\lambda)d\lambda$, where $...
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Summing Over General Functions of two Primes
According to the aforementioned here, it is possible to replace a sum on a function $ f $ of cousins such as the following:
$$
\sum_{p\le x}f(p)=\int_2^x f(t) d(\pi(t))\tag{1}
$$
and further
$$
\int_2^...
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Right-Continuity of Integrator in Lebesgue-Stieltjes Integral
Consider a (Lebesgue-Stieltjes) integrable function $f:\mathbb R\rightarrow\mathbb R$ and a right-continuous function $F:\mathbb R\rightarrow\mathbb R$ of bounded variation. The goal is to compute the ...
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On the rate of convergence of $\log(\|f\|^p_p)$ as $p\rightarrow\infty$
Throughout this posting $(X,\mathcal{F},\mu)$ is a probability space and $f\in L_\infty(\mu)\setminus\{0\}$.
It is well known that $f\in L_p(\mu)$ for all $0<p<\infty$, $\phi(p)=\|f\|^p_p$ is ...
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Layer Cake Representation without Fubini (Exercise 1.6 of Lieb and Loss)
I would like a hint in the right direction for the following problem:
Lieb and Loss' Analysis defines the general Lebesgue integral as a special case of the layer cake representation: for a non-...
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If $p>1$, $u\in L^p ([0,1])$ is nonnegative, and $\int_{\{u\geq t\}} udx\leq \min\{1,t^{-p}\}$ for all $t>0$, prove that $\int_0^1 u^p dx \leq p$.
Problem $3$ from UCI's Fall 2024 Real Analysis Qualifying Exam reads as follows:
Fix $p>1$, and suppose that $u\in L^p([0,1])$ is nonnegative and that
$$\int_{\{u\geq t\}}u\hspace{.1cm}dx\leq \min\...
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Is $g(t) = \int_\mathbb{R} f(x)e^{-t^2x^2} dx$ differentiable?
Let $f$ be a real-valued Lebesgue integrable function, and define
\begin{equation}
g(t) = \int_\mathbb{R} f(x)e^{-t^2x^2}\, dx
\end{equation}
Is $g(t)$ differentiable for $t> 0$ ?
In my thinking, ...
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Summable derivative implies absolutely continuous
Theorem 7.21 in 'Real and Complex Analysis' of Rudin says: '' If $f:[a,b]\longrightarrow \mathbb{R}$ is a derivable function and if $f'\in L^1([a,b])$, then $f$ is absolutely continuous ''
7.21 ...
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Absolute integrability wrt. Lebesgue under change of variables
Assume we have a function $f \in L_1(\mathbb{R}^d)$ which is absolutely integrable wrt. Lebesgue on $\mathbb{R}^d$. Now for all matrices $A \in \mathbb{R}^{d \times f}$ with full column rank $f$ (and $...
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Absolute continuity almost everywhere and Radon Nikodym derivative
Let $(\mathcal{X}, \mathcal{F})$ and $(\mathcal{Y}, \mathcal{G})$ be measure spaces.
Let $\mu \ll \nu$ measures on $\mathcal{X}$ and $K:\mathcal{X}\times\mathcal{G}\rightarrow[0,1]$ a Markov kernel. ...
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0
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79
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Does there exist a measure on $[0,1]$ representing disintegration over a partition indexed by $[0,1]$?
Consider a measure space $(X,A,\mu)$, where $\mu$ is a probability measure. Moreover let $X=\bigcup_{p\in[0,1]}X_p$ be a disjoint union of measureable sets, i.e. $X_p\in A$ and for each $p,q \in[0,1]$ ...
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Show $\delta_{\prod(\mu,\nu)}(\pi) = \sup_{\varphi,\psi} [\int \varphi d\mu + \int \psi d\nu - \int \varphi d\pi - \int \psi d\pi]$
We let $X \sim \mu$ and $Y \sim \nu$ be the source space and target space for optimal transport. And we let $c: X \times Y \rightarrow \mathbb{R} \cup \{\infty\}$ be a cost function. We recall that $\...
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How to prove that $\sum^{\infty }_{m=1} \frac{m}{2^{n}} \mu (f\in [\frac{m}{2^{n}} \leq f<\frac{m+1}{2^{n}} )\uparrow \int f\ d\mu $ for $f\geq 0$
Let $A_{m,n}=\{ x:\frac{m}{2^{n}} \leq f(x)<\frac{m+1}{2^{n}} \} $ and $f\geq 0$. I can see that the sum is nondecreasing in $n$ since
$$\mu (A_{m,n})=\mu (A_{2m,n+1})+\mu (A_{2m+1,n+1})$$
Also $\...
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Evaluate a crazy limit (in the context of Probability)
At the moment I am taking a measure-theory based probability course. In the previous homework assignment we were asked the following:
Evaluate the limit $$\lim_{n\to\infty} \frac{1}{2^{n}} \int_{-1}^{...
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47
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A question regarding an integral estimate [closed]
Suppose $g$ is an integrable real-valued function defined on $\mathbb{R}^{n}$ and we have information/estimate about the Fourier transform of $g$. Let $f$ be a bounded real-valued function on $\...
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1
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Help proving some lebesgue Integral Identity
I'm reaching out because I'm genuinely stuck on a few questions from a homework assignment for my measure theory class. Despite spending the last couple of days trying to work through them, I keep ...
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Countable Additivity of Lebesgue Integrals
Let $h$ be a Borel measurable function such that $\int_{\Omega}hd\mu$ exists. Define $\lambda(B) = \int_Bhd\mu, B \in \mathcal{F}$. Then prove that $\lambda$ is countably additive on $\mathcal{F}$.
$$$...
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1
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171
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Another way to establish a Lebesgue change-of-variables formula
Would you please help me solve Problem 55 in Chapter 6 of Real Analysis, fifth edition, by Royden and Fitzpatrick, which I am self-studying. Chapter 6 deals with Lebesgue integration.
55. For $f$ and ...
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Homeomorphism / theorem that led to the spherical coordinate version of the following integral for the sphere's surface area?
In this question, one of the answers (starting with "Note that the parametrization of sphere is given by...") suggested to compute the surface area by doing the following:
With the ...
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