Questions tagged [fubini]

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Confusion on switching coordinate systems in double and triple integrals

Triple and double integrals seem very logical; however, switching from a rectangular integral to a cylindrical or spherical one seems a bit messy. The derivations for switching $dV$ from $dxdydz$ to $...
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65 views

Prove difficult identity using Fubini's theorem

$$ \lim_{n \to \infty} \frac{1}{n^{k-1}} \sum_{i_1 = 1}^n \ldots \sum_{i_k = 1}^n \frac{1}{i_1 + i_2 + \ldots + i_k + m} = \frac{k}{(k-1)!} \sum_{j=2}^k (-1)^{k-j} j^{k-2} {k-1 \choose j-1} \ln j $$ ...
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1answer
35 views

Understanding the double integral $\int_0^\infty\int_0^t f(x)g(t-x)dxdt$

I am dealing with the integral $\int_0^\infty\int_0^t f(x)g(t-x)dxdt$ and $g(x)=0$ if $x<0$. I need to arrive to $\int_0^\infty f(x)dx\int_0^\infty g(t)dt$. Using Fubini, I have: $$\int_0^\infty\...
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1answer
49 views

representation through special numbers

Let $n,r\in N$ and let $S(n,m)$ represent Stirling's number of the second kind. It is known that $\sum_{m=0}^n S(n,m)m!=F_n$ is a Fubini number. Is it possible to represent (or estimate from above) ...
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2answers
102 views

Why exactly can you change the order of integration in a double (and triple) integral?

I'm currently studying multivariable Calculus doing double and triple integrals, and I'm slightly confused on why one can change the order of integration for a double integral. I have my own ...
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30 views

Integral Identity on Measure Theory [duplicate]

I'm trying to prove the following identity: If $f:X \rightarrow \mathbb{R}$ is a bounded, continuous function, and $\mu$ is a borel measure in $X$, then $\int f d\mu = \int_0^1 \mu(\{x \in X ; f(x)>...
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45 views

Proving that the triple integral can be evaluated by iterated integrals

I am currently in a Calculus III class and we're learning triple integrals. While learning double integrals, I was introduced to Fubini's Theorem, stating the double integral is equal to the iterated ...
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2answers
52 views

Using Fubini to prove $p\int_t^\infty\frac1{s^{p+1}}(\int_0^s w(x) \, dx) \, ds=\int_t^\infty\frac{w(s)}{s^p} \, ds+\frac{\int_0^tw(x) \, dx}{t^p}$

It was written that using Fubini's Theorem for a positive function $w$ and $p>0$, the following relation holds \begin{equation*} p\int_t^\infty \frac{1}{s^{p+1}}\left( \int_0^s w(x) \, dx\right) \, ...
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61 views

Durret Example 1.7.4: problem guidance

I have three questions regarding this example in Durret: "Our last example shows that measurability is important or maybe that some of the axioms of set theory are not as innocent as they seem. ...
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55 views

Confusions with Fubini’s Theorem

Fubini's Theorem: Let $\mu,\nu$ be two $\sigma$-finite measures and $f$ be a measurable function. If $f \ge 0$ or $f \in L^1(\mu\times \nu)$, then $$ \int f ~d \mu \times \nu = \iint f ~d\mu ~d\nu = \...
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73 views

Does $\iiint_{\mathbb{R}^3} \frac{1}{1+x^2y^2z^2}$ converge?

Does $\iiint_{\mathbb{R}^3} \frac{1}{1+x^2y^2z^2}$ converge? My try from symmetry it's enought to look when $x,y,z\geq 0$ and changing variables to $u=xyz,v=y,t=z$ $\frac{\partial (u,v,t)}{\partial(x,...
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103 views

Finding $\iint_{A}\frac{dx\,dy}{(1+x^2)(1+x^2 y^2)}$ with Fubini's theorem

I have to solve the following double integral $$\iint_{A}\frac{dx\,dy}{(1+x^2)(1+x^2 y^2)}$$ with $A= \left[0,+\infty\right[ \times [0,1].$ So far I've tried to solve it integrating w.r.t. $y$ first. $...
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2answers
47 views

Calculate the volume of $D=\{(x,y,z):x^2+y^2+z^2<1,ax+by+cz<d\}$

Calculate the volume of $D=\{(x,y,z):x^2+y^2+z^2<1,ax+by+cz<d\}$ I've thought the with wolg a=b=0 and then try to calculate it but i think the integral this way becomes more difficult
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28 views

Evaluate $\iint(x^2+4y) \,dA$, where $R$ is bounded by $y=x$, $y=x^3$, $x\geq 0$

First I set the two $y$ values equal to each other. I got $x=x^3$ so $x$ must equal $0$ or $1$ to be true. Next I did the double integral evaluated from $0$ to $1$ as the outer integral and $x$ to $x^...
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1answer
61 views

Does $\iint_{\mathbb{R}^2}\frac{\sin(x^2+y^2)}{x^2+y^2}\,dx\,dy$ converge?

Consider $$\iint_{\mathbb{R}^2}\frac{\sin(x^2+y^2)}{x^2+y^2}\,dx\,dy$$ My try: changing to polar coordinates and then calculate the integral $$\int_{0}^{2\pi}\int_{0}^{\infty}\frac{\sin(r^2)}{r^2}\,dr\...
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71 views

Calculate the following integral $\iint_{T}\frac{x^2\sin(xy)}{y}\,dx\,dy$

Calculate the following integral $$\iint_{T}\frac{x^2\sin(xy)}{y}\,dx\,dy\,,$$ where $$T=\{(x,y)\in\mathbb{R}^2:x^2<y<2x^2,y^2<x<2y^2\}$$ I found the $1/2\leq x\leq 1,1/2\leq y\leq 1$ but ...
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1answer
39 views

How Lebesgue integration solved the problem of changing the order of integration will change the value of integration?

Our professor started a course in measure theory by stating the problems of Riemann integration. One of the problems he\she stated is the following double integration: $\int_{0}^{1}\int_{0}^{1} \frac{...
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1answer
18 views

Example of unequal iterated integral but that does not contradict Fubini's Theorem

Consider counting measure $\mu_1$ and $\mu_2$ on $X=Y=\mathbb{N}$ Define a function, $$ f(x,y) = 2-2^{-x} \ \text{if} \ \ x=y \\ \text{and}\\ f(x,y) = -2 + 2^{-x} \ \text{if} \ \ x=y+1 $$ I showed ...
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57 views

using Fubini's theorem to prove “second moment trick”

Reading about statistical inference on graphs, I came across what is called the "Second moment trick": Let $\Theta$ be a parameter space and for each $\theta \in \Theta$, let $\left\{P_{\...
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28 views

Stochastic Fubini

Provided a stochastic basis $(\Omega,\mathfrak A,\mathfrak F,P)$ supporting a continuous local martingale $M$ with time horizon $T>0$, and a measure space $(E,\mathfrak E,\mu)$, let $F: \mathbb [0,...
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1answer
31 views

About the remark 3 about Fubini's Theorem in “Calculus on Manifolds” by Michael Spivak on p.60

I am reading "Calculus on Manifolds" by Michael Spivak. On p.60, there is a remark(remark 3) about Fubini's theorem in this book. Spivak wrote: Since $\int_A \mathcal{L}$ remains unchanged ...
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1answer
47 views

Application of Tonelli-Fubini Theorem?

Let $(\Omega\text{, }\mathcal{F}\text{, }\mathbb{P})$ be a probability space and $\{B_t\}_{t\geq0}$ be a standard Brownian Motion. Consider $\text{Var}\left(\displaystyle \int_{0}^t B_sds\right)$. We ...
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34 views

fubini application on $\int_{0}^{1}\int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}}ydxdy$

I have the integral $$\int_{0}^{1}\int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}}ydxdy$$ I am trying to apply fubini theorem, so I am trying to changes the interval of $x$ and $y$ I got to the result that $0<...
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4answers
89 views

Calculate $\iint\frac{dxdy}{(1+x^2+y^2)^2}$ over a triangle

Calculate $$\iint\frac{dxdy}{(1+x^2+y^2)^2}$$ over the triangle $(0,0)$, $(2,0)$, $(1,\sqrt{3})$. So I tried changing to polar coordinates and I know that the angle is between $0$ and $\frac{\...
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1answer
65 views

Calculate $\iiiint_{x^2+y^2+u^2+v^2\leq 1}e^{x^2+y^2-u^2-v^2}\,dx\,dy\,du\,dv$

$$\iiiint_{x^2+y^2+u^2+v^2\leq 1}e^{x^2+y^2-u^2-v^2}\,dx\,dy\,du\,dv$$ So we just learned substitution and i thought maybe for this integral doing 2 polar subs for x,y and for u,v but i'm not sure ...
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1answer
32 views

Indicator and simple functions verify Fubini's Theorem (Lebesgue-Integral)

I'm working on a proof of Fubini's Theorem. The theorem says: Given $A\times B\in \mathcal{L}\times\mathcal{L}$ a Lebesgue measurable set in $\mathbb{R^2}$, and $f:A\times > B\rightarrow\...
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1answer
39 views

Finding $\iint _De^{\frac{x-y}{x+y}} dy\,dx$ where $D=\{(x,y):x\geq 0,y\geq 0,1\leq x+y\leq 2\}$

Integrate $e^{\frac{x-y}{x+y}} dy\,dx$ over the set $D=\{(x,y):x\geq 0,y\geq 0,1\leq x+y\leq 2\}$. I tried to do substitution $u=x-y$ and $v=x+y$ so i know that $ 1\leq v\leq 2$ but i couldn't figure ...
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29 views

Product of Counting Measure

Let $ M = (\mathbb{N}, 2^{\mathbb{N}}, c)$, where $c$ is the counting measure on the $\sigma$-algebra $2^{\mathbb{N}}$, i.e. the power set of the naturals. If I want to consider the measure space $\...
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1answer
15 views

Find Volume of parallelogram Spanned by Two Vectors using Fubini's theorem

The problem: Let $v,u\in \mathbb R^2$ be two vectors: $v=(v_1,v_2), u=(u_1,u_2)$, and consider the parallelogram P spanned by them (we also assume that all the points inside P $(x,y)$ are positive/...
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1answer
28 views

Double integral dosen't give same result using Fubini Theorem

I got the equation 0 < y $\le$ x < 1 $\iint k dxdy$ = 1 , where k is a constant but dosen't really matter ... My problem is that I've chose the domains of integration x $\in $ ( 0 , 1 ) and y ...
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1answer
36 views

Prove The Identity Using Tonelli's or Fubini's Theorem

Show the integral $\int_{0}^{\infty} \frac{e^{-y} - e^{-ry}}{y} dy = ln(r)$ by applying Tonelli's Theorem on the set $[0,\infty) \times [1,r]$. Well Im trying to apply Tonelli's theorem to prove ...
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25 views

Failure of Tonelli's theorem for non-measurable functions.

I was looking at wikipedia's counter-examples of Fubini/tonelli's theorem Suppose that $X$ is the first uncountable ordinal, with the finite measure where the measurable sets are either countable ...
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1answer
34 views

Counter-example to tonelli's theorem.

I don't understand why this example doesn't satisfy the conditions of the Tonelli result. It is said that the space are not $\sigma$-finite. But isn't the counting measure and the Lebesgue measure $\...
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Finding a convergent subsequence in $L^1(\mathbb{R})$ using a Fubini argument

I am new to solving non trivial-for me,at least- theorical problems,so I Have kind of general trouble, when constructing a proof, to see if all passages are correct. In particular, I tried to prove ...
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1answer
49 views

A squared integrals

Is it true that $$ \left(\int_{0}^{t}f(u)du\right)^{2}=\left(\int_{0}^{t}f(u)du\right)\cdot\left(\int_{0}^{t}f(v)dv\right)=\int_{0}^{t}\int_{0}^{t}f(u)f(v)dudv\;? $$ I have tried to prove it with ...
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44 views

Confusion about Fubini's theorem and when it does or doesn't apply

I've been reading into Fubini's theorem (rectangular regions) quite extensively over the last few days, and I've gathered the following: For Fubini's theorem to apply, the function must be Lebesgue ...
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39 views

Riemann integrable vs Lebesgue integrable for unbounded intervals

If not all Riemann integrable functions are Lebesgue integrable on unbounded intervals, e.g. $\int_{0}^{\infty} \frac{\sin{x}}{x}$, then would I be right in saying that Fubini's theorem wouldn't ...
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52 views

Conditions for Fubini's theorem in simple terms

Could someone explain the conditions required to satisfy Fubini's theorem in layman's terms, without going into measure theory? I'm a high school student trying to gain a simple understanding of when ...
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54 views

Failure of Fubini's theorem and the existence of the double integral

If the two iterated integrals of a function f(x,y) that we calculate under Fubini's theorem are different, then that means the integrals of the positive and negative parts of the function are both ...
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1answer
26 views

Fubini's Theorem for finding the expectation w.r.t measure

Let $X$ be a random variable, then for any $n\in \mathbb{N}, \mathbb{E}^\mathbb{P}(X^n)<\infty.$ Then use Fubini's theorem to prove that $$ \mathbb{E}^\mathbb{P}(X^n) = n \int_0^\infty t^{n-1}(1-...
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1answer
29 views

If a double integral does not converge absolutely, does that mean the function is discontinuous? [closed]

An assumption of Fubini's theorem is that: $$\displaystyle{\iint_{R} |f(x,y)| \thinspace \mathrm{d} A < \infty}$$ If this assumption is not met, and the integral of the absolute value of the ...
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1answer
118 views

How to use Fubini's Theorem to find a function $g(z)$

Use Fubini's Theorem to find a function $g(z)$ such that $\int_0^a\int_0^x\int_0^y f(z)\,dz\,dy\,dx=\int_0^a g(z)f(z)\,dz$ where $a$ is a real number. I tried to use Fubini's Theorem, $$\int_0^a\...
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1answer
49 views

(c) Calculate the homotopy fibre of the inclusion $i_{X} : X \vee X \rightarrow X \times X. $

Here is the question: Let $F$ be the homotopy fiber of the inclusion $X \rightarrow X \times X.$ (1)Show that $\pi_{i}(F) \cong \pi_{i +1}(X).$ Here is the answer of this part: Show that $\pi_{i}(...
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3answers
80 views

Evaluate $\sum_{r=1}^{\infty} \frac{1 \cdot 3 \cdot \ldots (2r-1)}{r!}\left(\frac{2}{5} \right)^{r}$

Evaluate $$\sum_{r=1}^{\infty} \frac{1 \cdot 3 \cdots (2r-1)}{r!}\left(\frac{2}{5} \right)^{r}$$ Let $$y=x + \frac{1 \cdot 3 \cdot}{2!} x^2 + \frac{1 \cdot 3 \cdot 5}{3!} x^3+\ldots$$ be the given ...
3
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1answer
66 views

Show that an infinite Integral equals an infinite sum

Show that $$\int_{0}^{+\infty} \frac{\text{sin} ax}{e^x-1} dx = \sum_{n=1}^{\infty} \frac{a}{n^2+a^2}$$ My classmate asked me this problem and I don't know how to prove it. I guess that it can be ...
2
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1answer
56 views

Proving something similar to $\mathbb E(g(X)) = \int_0^\infty g'(x) P(X>x) dP$

For a non-negative r.v. $X$ s.t. $\mathbb E{X}<\infty$, and $g$ s.t. $||g||_\infty$ and $||g'||_\infty$ are both $<\infty$ (where $||\cdot ||_\infty$ is the essential supremum), we have that $$\...
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1answer
70 views

An example of $\int_{X \times Y} f d(\mu \times \lambda )$ is finite but such that $\int_{Y} f_{x} d\lambda $ is infinite for some $x \in (0,1).$

Let $X = Y = (0,1)$ and let $ \mu = \lambda)$ be the Lebesgue measure on $(0,1).$ Find an example of a positive continuous function $f: (0,1) \times (0,1) \rightarrow \mathbb{R},$ such that $$\int_{X \...
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2answers
34 views

$\sum_{(s,t)\in S\times T} f(s,t)=\sum_{s\in S}\sum_{t\in T} f(s,t)$ via Fubini-Tonelli?

Let $f: S\times T \to [0,\infty[$ be a function. I wish to show that $$\sum_{(s,t)\in S\times T} f(s,t)=\sum_{s\in S}\sum_{t\in T} f(s,t)$$ I know that in general we have $$\int_X g d\mu =\sum_{x\...
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1answer
71 views

Why are we allowed to replace the integral with respect to the product measure $\mu$ with iterated integrals?(2)

My statement for Fubini theorem is: {Let $(X, \mathcal{S}, \mu)$ and $(Y, \mathcal{T}, \lambda)$ be $\sigma$-finite measure spaces. and let $f$ be a $\mathcal{S} \times \mathcal{T}-$measurable ...
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1answer
47 views

Why are we allowed to replace the integral with respect to the product measure $\mu$ with iterated integrals?

I was reading this question here Fubini's Theorem double integral with sin and $e^{-x}$ but I do not know why are we allowed to replace the integral with respect to the product measure $\mu$ ...