# Questions tagged [fubini-tonelli-theorems]

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

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### 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 ...
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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 ...
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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-...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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)-...
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### 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} ...
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### 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 ...
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### 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 ...
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 $\left\{ c \right\} \times \mathbb{R}^{n-1}$...