Questions tagged [fubini-tonelli-theorems]
For questions related to the Fubini-Tonelli theorem, a theorem for interchanging integrals.
329
questions
1
vote
0
answers
35
views
Stochastic Fubini and Hawkes process
I would like to prove the following equality, which seems to be a kind of stochastic Fubini.
Here $Z_t$ is a Poisson random measure on some subset of $\mathbb{R}^{2}_{+}$, $h : \mathbb{R}_{+}\to\...
- 1,272
1
vote
0
answers
30
views
Measurability of a two-argument function
Let $f$ be integrable on $\mathbb{R}^d$, let $E_\alpha : = \{x \in \mathbb{R}^d:|f(x)|>\alpha\}$. For a fixed $\alpha$, $E_\alpha$ is a measurable set.
Is the function $\chi_{E_{\alpha}}(x)$ ...
0
votes
1
answer
65
views
Solve $\int_0^1 \int_x^{\sqrt{x}} \exp \left(\frac{x}{y}\right) \mathrm{d} y \mathrm{~d} x$,
$\int_0^1 \int_x^{\sqrt{x}} \exp \left(\frac{x}{y}\right) \mathrm{d} y \mathrm{~d} x$
Can someone help with this integral?
I Tried Fubini but it doesnt work. i switched dx and dy then i integred over ...
- 47
0
votes
1
answer
46
views
Fubinis Theorem for Lebesgue Integrals
We have recently learned Fubinis Theorem for Lebesgue Integrals in lecture.
However, I do not see how it is all that useful.
Since in our formulation we consider a function $f$ from $ X \times Y \to [-...
- 83
1
vote
0
answers
51
views
Is a set zero-measured when the lower dimensional truncated set of which is zero-measured?
I have no clue about this question, so here are some questions that may have answers or counter-examples. I hope this leads to more relevant results. If you know how to deal with any of them, or know ...
- 57
0
votes
0
answers
16
views
Fubini theorem for sequences of random matrices
Fubini’s theorem for sequences says:
If $\sum_{n=1}^\infty \sum_{m=1}^\infty |a_{mn}|<\infty$,
then
$\sum_{m=1}^\infty \sum_{n=1}^\infty a_{mn}=\sum_{n=1}^\infty \sum_{m=1}^\infty |a_{mn}|$
My ...
- 113
0
votes
0
answers
40
views
Problem 3-26 in "Calculus on Manifolds" by Michael Spivak. How to solve this problem using Fubini's theorem effectively?
The following problem is Problem 3-26 in "Calculus on Manifolds" by Michael Spivak.
I could solve this problem directly from the definition of "integrable" and "Jordan-...
- 125
5
votes
1
answer
263
views
Maximal Inequality for Expectation of Random Variables
Let $Z_1, Z_2, ..., Z_n$ be non-negative random variables, not necessarily independent. Then, show the expectation of the product of the random variables is less than or equal to the integral of the ...
- 53
1
vote
1
answer
43
views
infinite sum of measures estimate
I got stuck solving the following problem:
Consider a measure space $(X, \Sigma, \mu)$ and a non-negative measurable function. Furthermore, suppose $f$ is bounded and integrable. Show that $$\sum_{k = ...
- 11
0
votes
1
answer
52
views
Resnick - Probability Path - Exercise 5.17
Resnick (A Probability Path, 5.10.17) asks to show the following formula for two distribution functions $F$ and $G$ with no common discontinuity points in the interval $(a,b]$:
$$\int_{(a,b]} G(x) F(...
2
votes
0
answers
52
views
Integrability of the Jacobi Theta Function
Let $$\psi(x) = \sum_{n = 1}^{\infty} e^{-n^{2} \pi x}$$ be a theta function. Can it be shown that that
$$\int_{0}^{\infty} \psi(x) \cdot dx < \infty$$ without invoking Fubini-Tonelli’s Theorem ...
1
vote
0
answers
27
views
Relationship between surface integral and fubini's theorem
I am studying about surface integral and recently have studied about fubini's theorem.
And I think there exists any relationship between 'surface integral' and 'fubini's theorem'.
As a result, I came ...
- 69
2
votes
1
answer
179
views
Continuity of the inner-product in a Hilbert space with respect to unordered sums
Let $\{x_\alpha\}_{\alpha\in A}$ be an indexed set of elements in a Hilbert space. We say that the unordered sum $\sum_{\alpha\in A}x_\alpha$ converges to $x$ if for every $\epsilon>0$ there is ...
- 163
1
vote
0
answers
100
views
A proof of Poincare inequality
I stuck when reading the following proof of the Poincare inequality (Calculus of variations, Jurgen Jost & Xianqing Li-jost, Page 177-178):
Theorem (Poincare inequality)
Let $\Omega\subset\Bbb R^...
- 121
4
votes
0
answers
40
views
Fubini-like statement, reference request
Let $(X,\Sigma_X,\mu)$ and $(Y,\Sigma_Y,\nu)$ be finite measure spaces (the measures are finite, not the sets).
Let $k:X\times\Sigma_Y\to[0,1]$ be a Markov kernel which disintegrates $\nu$, i.e. with ...
- 7,937
0
votes
0
answers
23
views
Fubini's Theorem Holds After Changing a Condition in the Hypothesis
Prove that the conclusion of Fubini's theorem remain in force if we replace the condition that $f$ is integrable by $\int\left(\int\left|f_x(y)\right| d \mu_2(y)\right) d \mu_1(y)<\infty$.
My ...
- 679
0
votes
0
answers
31
views
Does Fubini's theorem apply on this infinite region?
I came across the following example for a triple integral:
Find the volume of the region bounded by hyperbolic cylinders:
$$ xy = 1 \quad , \quad xy = 9$$
$$ xz = 4 \quad , \quad xz = 36 $$
$$ yz = 25 ...
- 43
0
votes
0
answers
39
views
Intgration by parts formula for $\varphi \in C_C^{1}(\mathbb R^n)$
Let $\varphi \in C_C^{1}(\mathbb R^n)$ show that
$$\int_{\mathbb R^n} \nabla \varphi(x) dx=0$$
i´m a litle confused with the notation $\nabla \varphi(x)$ because $\varphi$ is a scalar field, the hint ...
- 149
0
votes
0
answers
35
views
Expectation of a product of random variable : definition
I have an hard time to understand several things I see on the expectation of two random variables.
First, here is my definition of product of two real valued random variables $X$ and $Y$ defined on ...
- 1,272
1
vote
1
answer
94
views
How to properly switch the order of integration?
Consider a random variable $X$ in $\mathcal{L}^1(\Omega,F,P)$ with distribution function
$F(X,t)=P(X\leq t)$ for $t\in\mathbb{R}$.
Apparently, it is true that
$$\int_{-\infty}^\eta F(X,t)dt = \mathbb{...
- 47
1
vote
0
answers
48
views
Applying Fubini's theorem to find the expected value of a random variable
I don't understand how we get this equality:
With $y$ non negative,
$$\int_0^{\infty}\left(\int_y^{\infty}f(x)dx\right) dy$$
$$=\int_0^{\infty}\left(\int_0^{x}dy\right)f(x)dx$$
When I apply Fubini, I ...
- 419
1
vote
1
answer
195
views
+50
Fubini's theorem for transition kernels
In probability theorey there is a different form of Fubini's theorem that includes Markov kernels (regular conditional distributions) that does not need independence. Let $(\Omega, \mathcal{A}, \...
- 333
2
votes
0
answers
53
views
Limit of Integral involving a bounded unit periodic function [duplicate]
I am trying to show that if $u : \mathbb{R} \to \mathbb{R}$ is a bounded, measurable, periodic function with period $1$ and $g$ is Lebesgue integrable over $\mathbb{R}$ then
$$
\lim_{n \to \infty} \...
0
votes
0
answers
65
views
Does Fubini's theorem also apply for non-rectangular sets?
Fubini's theorem is usually stated as
$$\int_{X\times Y}f d(\mu \otimes \nu) = \int_X \int_Y f(x, y) d\mu(x) d\nu(y).$$
So it is always defined in terms of integrations over the whole spaces. I was ...
- 333
1
vote
1
answer
58
views
Fubini's theorem and symmetry arguments
I'm currently going a textbook in a physics course and in there is the following expression $$\mathbb{E}_{x}\left[\mathbb{E}_{y}\left[f(x,y)\right]\right],$$
where $y \overset{\mathrm{i.i.d}}{\sim} \...
1
vote
0
answers
44
views
Show the following function is well defined
Consider the sequence $f_n(x)=e^{-nx}-2e^{-2nx}$ for $x>0$
a)Show $f(x):=\sum^\infty_{n=1}f_n(x)$ is well defined for $x>0$ and determine $f$
b) Show $$\sum^\infty \int_{(0,\infty)}f_n(x)\neq \...
- 921
0
votes
2
answers
150
views
Application of Fubini's Theorem - showing an integral-defined operator on $L^p$ outputs measurable functions
I am trying to understand the solution to problem 79 from chapter 7 of Alberto Torchinsky's Problems in Real and Functional Analysis. The problem is:
Let $(X,\mathcal{M}, \mu)$ and $(Y,\mathcal{N},\...
- 63
3
votes
0
answers
43
views
Fubini theorem on non-$\Sigma$-finite measures
I have been studying Fubini theorem and its proof on "Probability and Stochastics" by Erhan Cinlar.
Premise: a measure $\mu$ on a measurable space $\big( E,\mathcal{E} \big)$ is said to be $\...
- 311
0
votes
1
answer
58
views
Can I change the order of integration when the upper limits are infinite?
I'm trying to solve this :
$$\frac{d}{dy}\int_{y}^{\infty} \int_{f(y)}^{\infty} (g(y)+h(x_2))f(x_2)dx_2 f(x_1)dx_1$$
In this case, can I change the order of Integration ? I will get this :
$$\frac{d}{...
- 25
1
vote
1
answer
81
views
Show that $\sum_{(n,m)\in A\times B}x_{n,m}=\sum_{n\in A}\sum_{m\in B}x_{n,m}=\sum_{m\in B}\sum_{n\in A}x_{n,m}.$ Tao's Introduction to Measure Theory
I am reading "An Introduction to Measure Theory" by Terence Tao.
Exercise 0.0.2 (Tonelli's theorem for series over arbitrary sets). Let $A,B$ be sets (possible infinite or uncountable), and ...
- 8,510
0
votes
1
answer
43
views
Iterated integral where the density is not continous at the limit of integration
The following question is about measure theory
I was looking at the calculations of the integrals withing this post Why the Fubini theorem fail??
I was wondering about this equality $\int\int\frac{\...
- 515
13
votes
1
answer
441
views
Show $\int_0^{\pi/2}\int_0^1 e^{t+t^{\tan\theta}}dtd\theta=\frac{\pi}{4}(e^2-1)$
A friend gave me this double integral a while ago, and I couldn't figure out how to solve it.
$$\int_0^{\pi/2} \int_0^1 e^{t+t^{\tan\theta}}\,dt d\theta=\dfrac{\pi}{4}\left(e^2-1\right)$$
I tried ...
- 2,282
2
votes
1
answer
60
views
Equality of expected value using Fubini's theorem
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, $\alpha > 0$ and $X: \Omega \to \mathbb{R}$ a non-negative real-valued random variable. I need to prove that
$$\int_{[0,\infty)}\...
- 145
0
votes
0
answers
20
views
Clarification with regards to Fubini
In Wikipedia, https://en.wikipedia.org/wiki/Fubini%27s_theorem, it has been stated that:
The advantage of the Fubini–Tonelli over Fubini's theorem is that the
repeated integrals of $|f|$ may be ...
- 491
0
votes
0
answers
32
views
multivariate integral of a function defined by an inner product
Let $v=(v_1,...,v_d)$ be a vector in $\mathbb{R}^d$ and consider the aligned rectangle $R=[a_1,b_1]\times...\times[a_d,b_d]$ in $\mathbb{R}^d$. Calculate the integral $\int_{R}^{}e^{-\langle x,v\...
- 381
0
votes
0
answers
48
views
Double integral of $f(x,y)=\frac{xy}{(1-|x|)^2+(1-|y|)^2)}$ on $[-1,1] \times [-1,1]$
The problem is :
Is $f(x,y)=\frac{xy}{(1-|x|)^2+(1-|y|)^2}$ Lebesgue integrable on $[-1,1] \times [-1,1]$ ?
What is the value of $\int\int \frac{xy}{(1-|x|)^2+(1-|y|)^2} dydx $ and $\int\int \frac{xy}{...
- 103
0
votes
0
answers
16
views
Show that $E(Y)-E(X)=\int_{-\infty}^\infty[P(X<t\le Y)-P(Y<t\le X)]\lambda(dt)$ [duplicate]
If $X,Y$ are defined on the same probability space and have finite expectations, show that $$E(Y)-E(X)=\int_{-\infty}^\infty[P(X<t\le Y)-P(Y<t\le X)]\lambda(dt).$$
My attempt:
\begin{align}
E(Y)-...
- 1,989
0
votes
1
answer
62
views
Use of Fubini's theorem to check if equalities holds
I'm working on the following problem.
Would you check if my solution makes sense? I feel like I need more details in my argument. I don't know if it suffices.
Problem:
Let $D = \{(x, y) \in \mathbb{R}...
- 163
5
votes
1
answer
108
views
Why does Fubini's theorem not work in $\int_2^4\int_{x/2}^\sqrt{x}xy\ dy\ dx$?
$$\int_2^4\int_{x/2}^\sqrt{x}xy\ dy\ dx = \frac{11}{6}\neq\int_{x/2}^\sqrt{x}\int_2^4xy\ dx\ dy=3\left[x-\frac{x^2}{4}\right]$$
How do I know that Fubini's theorem ought not to work here?
- 3,165
1
vote
0
answers
91
views
Interchanging summations over sets
In order to understand two different definitions of the same function $f$ (as stated by Besner, 2022), I am trying to prove that those expressions are equal:
$\Delta(A) = f(A) - \sum_{B \subset A} \...
- 43
5
votes
1
answer
109
views
Convert $\frac1b\sum_{n=1}^\infty\frac{(b e^a)^n}{n!}B_{n-1}(an)$ to integral using $B_n(x)=\frac{n!}{2\pi i}\oint\frac{e^{x(e^t-1)}}{t^{n+1}}dt$
$\def\B{\operatorname B}$
In
How to solve $x^{y^z}=z$
A solution uses Bell polynomials $\B_n(x)$
$$e^{ae^{bz}}=z=1+\frac1b\sum_{n=1}^\infty \frac{(ae^b)^n}{nn!}\B_n(b n)=\frac1b\sum_{n=1}^\infty\...
- 10.6k
0
votes
0
answers
23
views
How do I take an expectation over a continuum of iid random variables?
short version: How do I take an expectation over a continuum of iid variables, i.e.:
$$\mathbb{E}_{\theta_{-i}} [U(\theta_i, \boldsymbol{\theta}_{-i})]$$ where $\theta_j$ is essentially a map $\theta: ...
- 21
0
votes
0
answers
101
views
Interchanging expectation and integration with a collection of random integrands
Suppose one has a collection of i.i.d. random functions $\{f(\cdot,t):t\in\mathbb R\}$, where we write $f$ for the common distribution. Assume that $f$ is integrable a.s., and that $\mathbb E\|f\|_1&...
- 3,013
0
votes
1
answer
103
views
How to use Hölder inequality to prove this integral inequality?
Consider an integral operator $Tf(x)=\int_{\mathbf{R}^n}K(x,y)f(y)dy.$ And $s,r \in(0,\infty), s \geq r$ are two indices. I would like to prove
\begin{equation}
\| Tf\|_{r} \leq (\int_{\mathbf{R}^n} ...
- 371
0
votes
1
answer
117
views
Fubini's theorem in Lebesgue integral theory. Incomprehensible point in the proof by Axler.
I'm reading the book on measure theory by Axler, https://measure.axler.net/MIRA.pdf.
I'm trying to understand the proof of Fubini's theorem in p.$132$.
Fubini's theorem
Let $(X,\mathscr S,\mu),(Y,\...
- 3,232
0
votes
0
answers
20
views
directional integrals to surface integrals
I want to convert a integrals over two direction that are not orthogonal into a integrals over the area.
As shown in the figure 1, the target is to compute $\int_{AB}\int_{AC}f\mathrm{d}s_1\mathrm{d}...
1
vote
0
answers
43
views
Calculate $ \int \limits_{V}\left(x^{2}+y^{2}\right) d(x, y, z) $, where $V$ is limited by the areas $ x^{2}+y^{2}= $ $ 2 z $ and $ z=2 $.
Calculate the integral $ \int \limits_{V}\left(x^{2}+y^{2}\right) d(x, y, z) $, where $ V \subset \mathbb{ R}^{3} $ is limited by the areas $ x^{2}+y^{2}= $ $ 2 z $ and $ z=2 $.
One can see that the ...
- 139
0
votes
0
answers
21
views
Exchanging order of integration and summation on the inner product of the partial Fourier series of $L^2$ functions
Let $f,g\in L^2(\mathbb{T}^1\times\mathbb{T}^1)$, where $\mathbb{T}^1$ is the one dimensional torus (a.k.a $\mathbb{S}^1$), that is, $L^2$ and periodic in each variable. I'm trying to prove a partial ...
- 1,287
2
votes
1
answer
69
views
Lemma 6.2 Lee's Introduction to Smooth Manifolds
I am trying to understand how to apply Fubini's Theorem to the following lemma
Suppose $A \subset \mathbb{R}^n$ is a compact subset whose intersection with $\left\{ c \right\} \times \mathbb{R}^{n-1}$...
- 5,253
1
vote
1
answer
92
views
An Application of Fubini-Tonelli Theorem
In a question here, selected answer is started with an application of Fubini's theorem. which I can't figure it out at all. I'm having difficulties with figuring out the differential elements. such as ...
