All Questions
32
questions
0
votes
2
answers
59
views
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. ...
- 1,221
0
votes
0
answers
54
views
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 ...
- 356
0
votes
0
answers
52
views
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-...
- 1,138
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 $$\sum_{k = ...
- 13
4
votes
0
answers
42
views
Fubini-like statement, reference request
Let $(X,\Sigma_X,\mu)$ and $(Y,\Sigma_Y,\nu)$ be finite measure spaces (the measures are finite, not the sets).
Let $k:X\times\Sigma_Y\to[0,1]$ be a Markov kernel which disintegrates $\nu$, i.e. with ...
- 8,127
3
votes
0
answers
74
views
Fubini theorem on non-$\Sigma$-finite measures
I have been studying Fubini theorem and its proof on "Probability and Stochastics" by Erhan Cinlar.
Premise: a measure $\mu$ on a measurable space $\big( E,\mathcal{E} \big)$ is said to be $\...
- 385
0
votes
1
answer
125
views
Use Tonelli's theorem to show an inequality resulting from the convolution of integrable functions.
Let $f$ and $g$ be Lebesgue integrable functions on $(\mathbb R, \mathcal B)$ to $\mathbb R$. If $\lambda$ denotes Lebesgue measure on $\mathcal B$, use Tonelli's theorem and the fact that
$$
\int_{\...
0
votes
1
answer
63
views
What does $\int_{\Omega_{1}} d \mu_{1} \int_{\Omega_{2}}|F(x, y)| d \mu_{2}$ mean in this Fubini's theorem?
I'm reading Chapter 4 in Brezis's Functional Analysis, Sobolev Spaces and Partial Differential Equations.
Let $\left(\Omega_{1}, \mathcal{M}_{1}, \mu_{1}\right)$ and $\left(\Omega_{2}, \mathcal{M}_{2}...
- 5,817
4
votes
1
answer
206
views
Why do I need Fubini to evaluate these integrals of products of functions?
I am currently reading a book on applied mathematics, and the current chapter is proving some statements related to probability theory. I'm stuck with one step in a derivation, because the author ...
- 93
0
votes
1
answer
51
views
Change of Integration Domains after Tonelli in Higher Dimensional Integrals
This might be considered as too general of a question but it has been bugging me for quite a while now: How does one change integration domains after using the Fubini-Tonelli's Theorem? Especially in ...
- 1,196
1
vote
1
answer
150
views
Proving a corollary of Lebesgue's Dominated Convergence Theorem
In reading Stein & Shakarchi's Real Analysis, I noticed that the authors apply the Dominated Convergence Theorem to not only sequences of functions $\{f_n\}_{n=1}^{\infty}$, but also to more ...
- 1,054
0
votes
1
answer
97
views
Double integration with $e^{-x^2}$
I am learning Fubini right now and I want to integrate
$$
\int_U e^{-x^2}y d\lambda_2 ,
$$
whereby
$$
U=\left\{(x, y) \in \mathbb{R}^{2}: 0 \leq y \leq 1, \quad y^{2} \leq x \leq 1\right\}
$$
But as ...
- 2,308
3
votes
1
answer
46
views
Fubini and induction for a sum over a set $Q$
How to calculate
$$
\int_{Q}\left(x_{1}+x_{2}+\ldots+x_{n}\right)^{2} d \lambda_{n}
$$
whereas $n \geq 2$ and
$$
Q=\left\{\left(x_{1}, \ldots, x_{n}\right) \in \mathbb{R}^{n}: 0 \leq x_{i} \leq 1, i=1,...
- 2,308
3
votes
0
answers
287
views
Fubini's theorem for conditional measures
I have an integration that looks like:
\begin{align}\label{eq1}\tag{1}
\int_{f \in F} \left[\int_{x \in \mathbb{R}} \chi_{\{x \in A\}} \mathrm{d} \gamma(x|f)\right] \mathrm{d} \mu(f),
\end{align}
...
1
vote
1
answer
28
views
Understanding tuple-indexed measures and integrating them
I have a measure $\mu $ that is supported on $[-3,3 ] \times \mathbb{R}$. What we are given is that, if we fix the first component $i$, then $\mu(i,\cdot)$ is a probability measure. Formally (maybe it ...