Questions tagged [fubini-tonelli-theorems]

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

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33 views

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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37 views

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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5 views

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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1answer
51 views

How do I compute the following integral with splitting the domain?

I have problems in computing the following integral $$\int_\Bbb{R} \int_\Bbb{R} |f(x,y)| dx dy$$ where $$f:\Bbb{R}^2\rightarrow \Bbb{R}; (x,y)\mapsto \frac{1}{x^2}\,\,\text{if}\,\,0<y<x<1,(x,...
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45 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 ...
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50 views

Applying Fubini's theorem for spherical coordinates

I want to prove that if $B^3_R$ is a 3-ball of radius $R$ around $0$, then $$ \int_{B^3_R}||x||^p dx = 4\pi\frac{R^{p+3}}{p+3}$$ I of want to use Fubini's theorem here to switch to spherical ...
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1answer
62 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 ...
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37 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,...
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33 views

How to prove Fubini-Tonelli for "Cavallieri" like sets?

One can sort of combine Cavalieri's principle and Fubini's theorem in a way that the following statement holds: $$ \int_T f(x,y) d(x,y) = \int_{\mathbb{R}} \int_{T_x} f(x,y) dx = \int_0^1 \int_0^{1-y} ...
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Converging function with dense set of singularities

Since my question here seems to be too complicated (since even though I put a bounty on it nobody answered or commented) I want to break it down a little bit. Maybe it gets answerable by reducing it ...
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1answer
27 views

Inversion formula and characteristic functions for a point mass

Durett Probability Theory and Examples suggest that the following inversion result (p.95) is intuitive. However, I cannot figure out how to prove it. Here is the result : If $X$ has characteristic ...
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33 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} ...
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18 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 ...
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Relation between the Hilbert-Hankel operator and Laplace transform on $L^2(\mathbb R_{>0})$

Let $\mathbb R_{>0} = (0, \infty)$. The Hilbert-Hankel operator $H$ is the integral kernel operator on $L^2(\mathbb R_{>0})$ defined as $$(Hf)(x) = \int_0^\infty \frac{f(y)}{x+y}\, dy$$ Prove ...
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Prove $\int_{R^1}{\frac{\delta^{\lambda}(y)f(y)}{|x-y|^{1+\lambda}}\,dy}$ is integrable and finite almost everywhere.

Let $F$ be a closed subset of $R^1$ and let $δ(x) = δ(x, F)$ be the corresponding distance function. If $λ > 0$ and $f$ is nonnegative and integrable over the complement of $F$, prove that the ...
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88 views

The Laplace transform $\mathcal L$ is a self-adjoint operator on $L^2(\mathbb R_+)$

Let $\mathbb R_+ = [0, \infty)$ and consider the kernel $K(x,y) = e^{-xy}$ on $\mathbb R_+ \times \mathbb R_+$. The associated integral operator on $L^2(\mathbb R_+)$ is called the Laplace transform $\...
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74 views

$(A_K^*f)(x) = \int_0^1 K^*(x,y) f(y)\, dy$ for all $f\in L^2[0,1]$ - Fubini's theorem?

Let $K$ be a square-integrable kernel on $[0,1] \times [0,1]$, i.e. $$\int_0^1 \int_0^1 |K(x,y)|^2\, dx\, dy < \infty$$ and let $A_K$ be the integral operator induced by it on $L^2[0,1]$, i.e. $$(...
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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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91 views

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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1answer
65 views

How do I compute this integrals and conclude that Fubinis theorem doesn't hold everywhere?

I have the following problem: Let $\Omega_1=\Omega_2=\mathbb{N}$, $A_1=A_2$ the $\sigma$-algebra of all subsets of $\mathbb{N}$ $\mu_1,\mu_2$ the counting measure. We consider the function $$f:\...
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63 views

How do I compute this integrals?

I have the following question: Let $\Omega_1=\Omega_2=[0,1]$, $A_1,A_2$ the borel $\sigma$-algebra on $[0,1]$, $\mu_1$ is the lebesgue measure and $\mu_2$ the counting measure.Let $$C=\{(x,x)|x\in [0,...
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55 views

How do I compute this integral using the Fubini and/or Tonelli theorem?

I have the following problem: Using Fubini and/or Tonelli theorems, compute the Lebesgue integral: $$\int_{[0,1]} \left(\int_{[y,1]} x^{-\frac{3}{2}} \cos\left(\frac{\pi y}{2x}\right) \,\,d\lambda(x)\...
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1answer
36 views

Fourier transform of $ |x|^{\alpha - n}$

I am attempting to understand the following proof (Theorem 5.9) from Lieb-Loss. They remark that by Fubini $$ \int_0^\infty\lambda^{-n/2}\lambda^{\alpha/2-1}\left (\int_{R^n}\exp[-\pi|x-y|²/\lambda]f(...
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1answer
47 views

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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Kalman Filter with Non-Constant Acceleration

I want to model my state space using the following model: $$ a_{t} = a_{0} e^{-\alpha t} + \sqrt{2 \alpha \sigma^2} \int_{0}^{t} e^{-\alpha(t-s)} d W_{s} $$ $$ v_{t} = v_{0} + \int_{0}^{t} a_{s} ds ...
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107 views

Why do we need Fubini's theorem in this proof of Minkowski's inequality for integrals

I'm reading Theorem 6.19 in textbook Real Analysis: Modern Techniques and Their Applications by Gerald B. Folland. Suppose that $(X, \mathcal{M}, \mu)$ and $(Y, \mathcal{N}, \nu)$ are $\sigma$-finite ...
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2answers
54 views

A generalized version of Hölder's inequality

I've found this version from this Wikipedia page. I've re-written the proof to make my understanding clear. Could you confirm if my attempt is correct? Let $(X, \mathcal A, \mu)$ be a $\sigma$-...
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35 views

Fubini's theorem with unbounded but summable integrand

I am studying the function \begin{align*} %\int_{-\infty}^\infty x \rightarrow \int_{-\infty}^\infty \mathrm{arcsch} (|x-y|) \frac{2y^2+by-1}{\sqrt{1-y^2}} \mathbf{1}_{(-1,1)} \mathrm{...
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1answer
50 views

Why do we need hypothesis of complete measure in this version of Fubini's theorem?

I'm reading below Fubini's theorem in page 3 of this lecture note. Let $(X, \mathcal{A}, \mu)$ and $(Y, \mathcal{B}, \nu)$ be complete measure spaces, let $\gamma$ be the product outer measure on $X \...
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226 views

A detailed and self-contained proof of Tonelli's theorem

Motivation: I have seen the interchange of limit/derivative and integral many times, but don't know how such operation makes sense. I've always desired to remove this uncertainty by giving a ...
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55 views

Construct a collection of pairwise disjoint measurable sets from a set of rectangles

In proving the Fubini's theorem, I've come across below statement for which I'm not sure if it's indeed correct. Could you please confirm the validity of this statement and if my proof is correct? ...
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1answer
19 views

When can the Fourier transform change order in the inner product of $L^2? $

In $L^2(\mathbb{R}^n)$. Let $(f,g):=\int fg$. If $f\in\mathcal{S}(\mathbb{R}^n)$ and $g\in L^2(\mathbb{R}^n)$. When $(\mathcal{F}^{-1}(f),g)=(f,\mathcal{F}(g))$? This always holds in this case? Thanks....
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28 views

Non-finite integrals in my proof of Tonelli's theorem for characteristic functions

I'm trying to prove Tonelli's theorem for characteristic functions. Let $(X, \mathcal A, \mu)$, and $(Y, \mathcal B, \nu)$ be $\sigma$-finite measure spaces. $\Sigma = \{A \times B \mid A \in \...
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2answers
73 views

Volume of a subset of R^n

Question: Compute volume of the set A: $A = \{(x_1,x_2,...,x_n) \text{ | } 0 ≤ x_i ≤ 1, \sum_ix_i = 1\}$ Try: We know volume is basically integrating 1 over the set. I suppose fubini’s theorem is to ...
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1answer
69 views

Is this a general counterexample about Fubini's Theorem?

Let $\chi$ be a smooth and null-average function with compact support in $\mathbb{R}$. Let $f\in L^1_{loc}(\mathbb{R})$. The following iterated integral is well-defined $$ \int_{x \in \mathbb{R}} f (x)...
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1answer
66 views

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 "...
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39 views

How to compute the joint law of a Markov chain?

Let $(Y_n)_{n\ge 0}$ be a Markov chain with values in a measurable space $(E,\mathcal{E})$ and initial law $\nu=\text{Law}(Y_0)$. My lecture notes now claim the following regarding the law of $(Y_0,\...
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28 views

Limit of $\frac{1}{\epsilon}\int_{D(\epsilon)}\frac{1+\sin(x)+\sin(y)}{\sqrt{x^2+y^2}}\text{d}(x,y)$ over disk of radius $\epsilon$. [duplicate]

I have been bashing my head over the following problem: Define the closed disk of radius $\epsilon$ as $D(\epsilon)=\{(x,y)\in\mathbb{R}^2\hspace{2mm}|\hspace{2mm}0 < x^2+y^2\leq\epsilon^2 \}$. ...
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1answer
67 views

Integrating with the counting measure

I have found the following example in a book, where the example emphasizes the importance of the fact, that the $\left(X,\mathscr{A},\mu_{1}\right)$ and $\left(Y,\mathscr{B},\mu_{2}\right)$ are have ...
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38 views

Why is a rectangle $A \times B$ with $A \subset \mathcal{A}$, $B \in \mathcal{B}$ called a rectangle of $\mathcal{A} \times \mathcal{B}$?

In Freidman's Foundations of Modern Analysis, ch. 2.15 Product of Measures, he defines: The Cartesian product $X \times Y$ as all the ordered pairs $(x,y)$, where $x \in X$ and $y \in Y$. Rectangles ...
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1answer
26 views

Fubini theorem on integral with brownian motion

Consider $X_t = \int_0^t \int_0^t B_s B_r ds dr$ Computing $E[X_t]$ yields swapping integrals: $$E[X_t] = \int_0^t \int_0^t E[B_s B_r] ds dr $$ Why am I allowed to do this using Fubini's thereom? I do ...
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37 views

Does this change of order of integrals look right

A paper I'm reading has the following statement: Changing the order of integrals, $\int_{\epsilon}^{1-\epsilon}\left(\int_{0}^{x} G(t) d t\right) d \mu_{1}^{\star}(x)=\int_{0}^{1}\left(\int_{\epsilon}^...
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1answer
85 views

Fubini's theorem cannot be applied

I've been trying to tackle the following question: Consider the function $f:\mathbb{R}^2\to\mathbb{R}$ defined by $f=2\cdot 1_M-3\cdot 1_N$, where $$ M=\{(x,y):y\geq 0,x-2\leq y<x-1\}\;\text{and}\; ...
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1answer
68 views

Find all values of $p$, with $1\leq p \leq \infty $, for which $f\in L^p(X,\mu)$.

I am struggling on this qualifying exam question that I found. A hint is provided that says Fubini may be helpful here, but I can't see how to setup the problem to apply it. Here is the question: Let $...
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1answer
70 views

Recursion formula for volume of an unit ball in $\mathbb{R^n}$

Let $B^n\subset \mathbb{R^n}$ be an unit ball in $\mathbb{R^n}$ and $V_n(A)$ be the volume of $A\subset \mathbb{R^n}$.Then, prove \begin{equation} V_n(B^n)=2 V_{n-1} (B^{n-1}) \displaystyle\int_0^1 (...
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1answer
75 views

Fubini's theorem on locally compact Hausdorff spaces without measure theory

Suppose that $X$ is a locally compact Hausdorff space. For a Radon measure $\mu$ on $X$, let $I_{\mu}\colon C_{c}(X)\to\mathbb{C}$ be the positive linear functional defined by $I_{\mu}(f):=\int_{X}f \ ...
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1answer
51 views

Upper triangular matrix and Lebesgue measure.

Let $M$ be a $n\times n$ upper triangular matrix s.t. diagonal components are $1$. $\Bigg($For example, $M=\begin{pmatrix} 1 & 3 & 2 \\ 0 & 1 & 5 \\ 0 & 0 & 1 \end{pmatrix}, \...
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22 views

Reference for Tensor product of finitely additive probability measures on discrete space

Let us consider two discrete space $X$ and $Y$ with the corresponding $\sigma$-algebras $\mathcal{P}(X)$ and $\mathcal{P}(Y)$, respectively (where the notation $\mathcal{P}(X)$ means the power set of $...
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1answer
61 views

Finding limit of Double Integral

I'm working through some old qual problems and ran into this one that has stumped me. The problem is to compute $$\lim_{n\to \infty} \int_0^\infty \int_0^\infty \frac{n}{x} \sin\left( \frac{x}{ny} \...

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