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Questions tagged [fubini-tonelli-theorems]

For questions related to the Fubini-Tonelli theorem, a theorem for interchanging integrals.

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Linearity of expectation for infinite sums of positive random variables

I have a question regarding the linearity of the expectation of the infinite sum of positive random variables, namely I want to prove on a countable space $\Omega$ that \begin{equation} \mathbb{E}[\...
Leoncino's user avatar
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Failure of Fubini when integrating in the sense of distributions

Similar questions have been asked, but I find it hard to apply it to this case. The following is a physics motivated problem and should illustrate how Fubini fails. The main question is then: How do I ...
Confuse-ray30's user avatar
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Associativity of Convolutions

In Folland's real analysis textbook, there are the following propositions: Assuming that all integrals in question exist, we have $$ (f*g)*h=f*(g*h) $$ The proof is based on the Fubini's theorem.But ...
12345's user avatar
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How is this use of Fubini's theorem justified?

In the paper "A Comparison Theorem on Moment Inequalities Between Negatively Associated and Independent Random Variables" (Qi-Man Shao, 2000) the following theorem is proved (paraphrased): ...
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Need help understanding the switching of integral limits with Fubini's theorem: $\int_0^cm(\{x:|f(x)|>t\})dt=\int_{\{x:c\geq |f(x)|\geq 0\}}|f(x)|dx$

Let $f\in L^1(\mathbb{R}, m)$ with $m$ being the Lebesgue measure and $c > 0$. I have to admit that I am quite bad with using Fubini's theorem outside of calculus type problems of abstract proofs. ...
Cartesian Bear's user avatar
2 votes
1 answer
107 views

Applying a summation method to two sums. I need to justify an interchange of summation and integration with this method. Is my use of Fubini flawed?

Below I obtain the Leibniz formula for $\pi$ using a particular summation method. However, you need to justify a step where you interchange a summation and integration. Fubini would not work here. So ...
Dave77's user avatar
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Can the measurability assumption in Fubini's Theorem be relaxed?

If $(X, \mathcal{A}, \mu)$ and $(Y, \mathcal{B}, \nu)$ are $\sigma$-finite measure spaces, and $f: X \times Y \rightarrow \mathbb{R}$ is a function such that the iterated integrals $\int_X \int_Y f(x, ...
ZENG's user avatar
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4 votes
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Reversed Fubini's

Assume we have a real valued function $F:\mathbb{R}^{n} \times (0, \infty) \to \mathbb{R}$. And assume that we have the function $ g: \mathbb{R}^{n} \to \mathbb{R} $ given by $$ g(x) = \int_{0}^{\...
User091099's user avatar
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Apply Fubini's theorem to $\int^1_0\int^2_{y^2}x^2y-y^2x$

Can Fubini's theorem be used for the integral $$\int^1_0\int^2_{y^2}(x^2y-y^2x)\text{d}x\text{d}y\ \ ?$$ Why? If yes, explicitly write the integral with the corresponding integral limits. The ...
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General formula for reversing double integral bounds

The double integral over the region: $$ R = \left\{ \left( x,\: y \right) : a \leqslant x \leqslant b,\: g\left( x \right) \leqslant y \leqslant h\left( x \right) \right\} $$ is expressed as $$ \...
LightninBolt74's user avatar
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Is there a relation between Fubini's theorem and change of variable theorem?

In an exercise, it asks to use the change of variable theorem to calculate a double integral, but then it asks to redo the work using Fubini's theorem. Is there a way to benefit from previous work? In ...
Alia's user avatar
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On the hypothesis of Fubini's theorem

I'm reading Bauer's Measure and Integration Theory. After the proof of Fubini's theorem in page 140 he introduces the following example: I don´t understand why the theorem doesn't apply in this case. ...
Kham Bodrogi's user avatar
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Does Fubini theorem apply in this case?

I was trying to do some self-studying and learning more about Fubini or Tonelli's theorem, and I came across this problem maybe someone can help me with (excuse me if this is too trivial, I am very ...
stxsTIC's user avatar
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Fubini's theorem for Bochner Integral

I've just been (as of two days ago) introduced to the Bochner integral, and I've read that Fubini's theorem holds for it, but I haven't been able to find its version for the said integral. So here's ...
Gustavo de Souza's user avatar
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Why is uniform convergence needed in this proof of Leibniz Integral Rule?

Wikipedia's proof of Leibiz Integral Theorem begins as follows: We use Fubini's theorem to change the order of integration. For every $x$ and $h$, such that $h > 0$ and both $x$ and $x+h$ are ...
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Calculate a integral of : $I = \int \int_{V} y^{-2} d \lambda_2(x,y)$ where: $V = \{ x^2 + y^2 > 1, |x| < \frac{1}{2}, y > 0 \}$

Calculate a integral of : $$I = \int \int_{V} y^{-2} d \lambda_2(x,y)$$ where: $V = \{ x^2 + y^2 > 1, |x| < \frac{1}{2}, y > 0 \}$ Using help provided by @geetha290krm I was able to write ...
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measurability of a random variable taking values in the space of continuous functions

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability sapce and let $X: \Omega \to C^+(\mathbb{R})$ be a random variable taking values in the space of real valued non-negative continuous functions ...
mathematico's user avatar
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Let $I(x) := \int_{B(x,R)} \frac{f(y)}{|x-y|^{n-1}} dm(y)$. Then there exists $C = C(n)$ such that $|I(x)| \leq CR \cdot Mf(x)$

Let $\lambda_n$ be the $n$-dimensional Lebesgue measure, and let $f\in L^1_{loc}(\mathbb R^n)$ and $Mf$ be the maximal function of $f$. For fixed $R>0$ and $x \in \mathbb R^n$ and we define $$ I(x)...
Squirrel-Power's user avatar
4 votes
1 answer
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Frullani's integral

I know that there are a lot of posts that derive Frulani's integral but I would really like to know how to do it for myself. Here is the problem: Lef $f:[0,\infty)\to\mathbb{R}$ a $C^{1}$ function ...
Carlos Jiménez's user avatar
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Proving a given metric is a metric on the product of a measure space with itself.

Let $\mathbf{M}_1(X)$ denote the set of Borel probability measures on a compact metric space $X$. Given $\mu_1,\mu_2 \in \mathbf{M}_1(X)$, let $J(\mu_1,\mu_2)$ be the set of Borel probability measures ...
homosapien's user avatar
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There are 12 students. The student forms 6 groups to do a project. Each week, the students can form the group as the wish. Prove the following.

The full question is, There are 12 students in Mr. Fat's combinatorics class. At the beginning of each week, Mr. Fat assigns a project to his students. The students form six groups. Each group works ...
SuperMaxAli's user avatar
1 vote
0 answers
70 views

Looking for an example for Fubini's theorem

I am preparing a lecture about Fubini's theorem. For me, in "real life" the most common application of Fubini's theorem is to "change the order of almost-everywhere quantifiers". I....
the_lar's user avatar
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Does $\int_{[0,1]} f(x,y)\,dx = 0$ imply $\int_{[0,1]^2} f = 0$? [duplicate]

I always believed that $\int_{[0,1]} f(x,y)\,dx = 0$ (for all $y$) implies $\int_{[0,1]^2} f = 0$ by Fubini-Tonelli. However, in this upvoted answer here it says the opposite. So in particular, you ...
LordOfNumbers's user avatar
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Measurability of $u(x-y):(\mathbb{R}^n\times \mathbb{R}^n, \sigma(\mathscr{L}^n\times\mathscr{L}^n)) \rightarrow (R,\mathscr{B}(\mathbb{R}))$

When we define the convolution on $L^1(\mathbb{R}^n)$, we are interested to proof that $\forall f,g \in L^1(\mathbb{R}^n)$ then $f\star g \in L^1(\mathbb{R}^n)$. In the proof of this we want use the ...
Manuel Bonanno's user avatar
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Is it possible to demostrate, supported by the Fubini-Tonelli that integrate $h(x)(f*g)(x)$ is equivalent to integrate $f(x)(h*g)(x)$ [closed]

there is a lot of post about integral of the convolution of two fuction. Is it possible to demostrate that using variable changes as in the Fubini the following equivalence is valid: \begin{align}\...
Heberley Tobon Maya's user avatar
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$h(D\times\{x^n\})\subset\mathbb{R}^n$ and the author wrote $\int_{h(D\times\{x^n\})} 1 dx^1\cdots dx^{n-1}$. Is this ok? Calculus on Manifolds Spivak

I am reading "Calculus on Manifolds" by Michael Spivak. In the proof of Theorem 3-13 (Change of variable), the author wrote as follows: Let $W\subset U$ be a rectangle of the form $D\times [...
佐武五郎's user avatar
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Calculate a difficult integral with the help of the Theorem of Fubini

How would you possibly calculate this integral? $$ \int\limits_0^\infty e^{-y} \cdot \frac{ (\sin(y) )^2 }{ y } dy $$ I suppose that the solution involves applying the Theorem of Fubini. I attach the ...
user172501's user avatar
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1 answer
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Fubini - counterexample

On $[0,1]\times [0,\infty)$, consider the function $$ f(x,y)=(2-xy)xy\exp(-xy). $$ Why is Fubini's theorem not applicable? I get that $\int_0^\infty\int_0^1 f(x,y)\, dx\, dy =1\neq 0=\int_0^1\int_0^\...
selector's user avatar
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1 answer
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Problem 3-35(a) in "Calculus on Manifolds" by Michael Spivak. If I show $\chi_{g(U)}:C\to\mathbb{R}$ is integrable, then is my proof ok?

Problem 3-35 (a) Let $g:\mathbb{R}^n\to\mathbb{R}^n$ be a linear transformation of one of the following types: $$\begin{cases} g(e_i)=e_i & i\neq j \\ g(e_j)=ae_j & \end{cases}$$ $$\begin{...
佐武五郎's user avatar
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2 votes
0 answers
104 views

Problem 3-32 in "Calculus on Manifolds" by Michael Spivak. What are "considerably weaker hypotheses"?

Problem 3-32. Let $f:[a,b]\times [c,d]\to\mathbb{R}$ be continuous and suppose $D_2f$ is continuous. Define $F(y)=\int_a^b f(x,y) dx$. Prove Leibnitz's rule: $F'(y)=\int_a^b D_2f(x,y) dx$. Hint: $F(y)=...
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0 answers
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Switching the order of a triple integral and differing results, a Multivariable Calculus exercise

Here is a triple integral a student asked me, but which I can't seem to get right. I have tried to recalculate the integrals several times, but to no avail. If you're taking a Multivariable Calculus ...
tzy's user avatar
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$\int_D 1$, where $D:=\{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 = r^2, (r, z) \in C\}$. (Problem 3-29 in "Calculus on Manifolds" by Michael Spivak)

The following problem is a problem in the section "FUBINI'S THEOREM". Problem 3-29. Use Fubini's theorem to derive an expression for the volume of a set of $\mathbb{R}^3$ obtained by ...
佐武五郎's user avatar
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5 votes
1 answer
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Can you critique my exposition of $\int_{-\infty}^{+\infty}e^{-x^2}dx = \sqrt \pi$?

Prove $$\int_{-\infty}^{+\infty}e^{-x^2}dx = \sqrt \pi$$ using Fubini's Theorem. My solution is below. Proof is Correct. What I want to know is : Is it well written? How could the writing be improved, ...
SRobertJames's user avatar
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6 votes
1 answer
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Is Fubini’s theorem behind this equality?

I consider $H$ and $h$ two non negative functions. I had hard time to understand this equality $$ \int_{0}^{t}h(t-s)\left(\int_{0}^{s}H(s-u)udu\right)ds = \int_{0}^{t}u\left(\int_{u}^{t}h(t-s)H(s-u)ds\...
G2MWF's user avatar
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1 vote
0 answers
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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 $\chi_{E_{\alpha}}(x)$ ...
Noppawee Apichonpongpan's user avatar
1 vote
1 answer
73 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 ...
MathJason's user avatar
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1 answer
82 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 [-...
MathMaestro's user avatar
1 vote
0 answers
56 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 ...
daidaitx's user avatar
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0 answers
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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 ...
toni_iva's user avatar
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0 answers
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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-...
佐武五郎's user avatar
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5 votes
1 answer
309 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 ...
user29384756's user avatar
1 vote
1 answer
51 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 = ...
PesFAs2's user avatar
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1 answer
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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(...
picklechu's user avatar
2 votes
0 answers
66 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 ...
Robert Abramovic's user avatar
1 vote
0 answers
45 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 ...
KHJ's user avatar
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3 votes
1 answer
254 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 ...
Cat Branchman's user avatar
1 vote
0 answers
189 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^...
QIRUN CONG's user avatar
4 votes
0 answers
42 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 ...
geodude's user avatar
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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 ...
Vinay Deshpande's user avatar
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0 answers
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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 ...
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