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}[\...
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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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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, ...
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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
$$
g(x) = \int_{0}^{\...
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Apply Fubini's theorem to $\int^1_0\int^2_{y^2}x^2y-y^2x$
Can Fubini's theorem be used for the integral $$\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\}
$$
is expressed as
$$
\...
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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 ...
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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. ...
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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 ...
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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 ...
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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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