Questions tagged [fubini-tonelli-theorems]

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

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Double integral with the Dirac's delta

Problem For a given $T\in\mathbb{R}$ and $K\in\mathbb{N}$, consider the following sequence of points \begin{equation*}t_k\triangleq kT \qquad k=0,1,\dots,K\end{equation*} I need to compute the ...
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Question on Fubini's theorem [closed]

$$\displaystyle \int_a^b\varphi'(x)\left(\int_a^xf(t)dt\right)=\int_a^b\left(\int_t^b\varphi'(x)f(t)dx\right)dt$$ I do not understand why this equality holds true. Can someone please help me? Many ...
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Transformation of Area restricted by 3 functions and x-Axis

With the help of a suitable transformation and Fubini I want to determine the integral $$\int_{V} x^{3} y d \lambda_{2}(x, y),$$ where $V$ is the open subset of $\mathbb{R}_{+}^{2}$ bounded by the ...
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A detailed and self-contained proof of Fubini's theorem for Banach spaces

After so much preparation (in proving auxiliary lemmas), I finally complete the proof of Fubini's theorem for Banach spaces. This is what I have desired after proving Tonelli's theorem :) The journey ...
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Converging function with dense pole set

Let $$f: \mathbb C^\times \to \mathbb R, \quad f(z):= \begin{cases}\log(|z|),\quad&|z|<1,\\ 0, &|z| \ge 1, \end{cases},$$ and $(a_n)_{n \in \mathbb N}$ be a sequence of positive real ...
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Let $f \in \mathcal S (X \times Y, \lambda, E)$ and $f_x:y \mapsto f(x, y)$. Then $\Phi : x \mapsto [f_x]$ is integrable

In generalizing Fubini's theorem to functions on Banach space, I encounter a result that I'm unable to prove. Could you shed some light on this issue? Related definitions of Bochner integrals can be ...
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Fubini's theorem on a flow

Suppose $f \in L^1(X)$ and let $\phi_s :X \to X$ be a one parameter measure preserving flow that is also bi-measurable on $R \times X$. Show that for almost every $x \in X$ the function $f(\phi_s(x))$...
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How do we reduce the inversion formula for the $n$-dimensional Fourier transform to the $1$-dimensional case using Fubini's theorem?

Please consider the following theorem from Measures, Integrals and Martingales (2nd edition): Question 1: The proof of the claim is straightforward, but I don't understand the remark in the "...