# 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 \mathbb{E}[\...
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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 ...
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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 ...
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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. ...
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### 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 ...
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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 ...
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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 ...
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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, ...
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