All Questions
70
questions
1
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149
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How to use the hint by the author? (Problem 3-23 in "Calculus on Manifolds" by Michael Spivak.)
Problem 3-23.
Let $C\subset A\times B$ be a set of content $0$. Let $A'\subset A$ be the set of all $x\in A$ such that $\{y\in B:(x,y)\in C\}$ is not of content $0$. Show that $A'$ is a set of measure ...
- 1,138
1
vote
0
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179
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Problem 3-21 in "Calculus on Manifolds" by Michael Spivak.
The following problem is Problem 3-21 in "Calculus on Manifolds" by Michael Spivak.
I solved this problem but I am not sure if my solution is right. Is my solution right? If so, how to ...
- 1,138
2
votes
1
answer
84
views
Rudin's "Real and Complex Analysis" Exercise 1.10 - Generalization
Exercise:
Suppose that $\mu(X) < +\infty$, $(f_n)_n$ is a sequence of bounded complex measurable functions on $X$ and $f_n \to f$ uniformly on $X$. Prove that:
$$ \lim _{n \to +\infty} \int _X f_n ...
- 1,179
2
votes
1
answer
81
views
Question about counting measure
I want to check if my solution is right. Can someone help me?
Consider the measure space $(\mathbb{R}, 2^{\mathbb{R}}, µ)$ where $µ$ is the counting measure, hence the measure of a set is its ...
- 900
2
votes
0
answers
78
views
Prove an Integral Function is Differentiable
Everything is under the Lebesgue integral setting. Fix $p \in \mathbf{R}$. Define for $y \in \mathbf{R}$,
$$
F(y) := \int_0 ^\infty \frac{\sin(xy)}{1 + x^p} \,dx.
$$
It can be proven that $F(y)$ is ...
- 1,196
2
votes
2
answers
79
views
Let $Y=h(X),$ Find $E\{Y\}$
Problem:
Let $X: (\Omega, \mathscr{A}) \rightarrow (\mathbb{R},B)$ be a random variable with the uniform distribution $P^X=\frac{1}{2\pi}\mathbb{1}_{\{(0,2\pi)\}}$ on the interval $(0,2\pi),$ and $h:(\...
- 416
2
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0
answers
59
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$\int_W x^2y^2 = \int_V x^2y^2$. Is my proof of this equation right? ("Analysis on Manifolds" by James R. Munkres)
I am reading "Analysis on Manifolds" by James R. Munkres.
The following example is EXAMPLE 4 on p.149:
EXAMPLE 4. Suppose we wish to integrate the same function $x^2y^2$ over the open set $$...
- 8,740
0
votes
0
answers
74
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Alternate Proof of the Monotone Convergence Theorem
I am following Folland’s Measure Theory book, and it contains a proof for the MCT that uses the fact that the integral of a simple function induces a measure, and here is my proof without that fact.
...
- 436
4
votes
1
answer
781
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Measure of Compact Set with Empty Interior
For my Integration course I've been proposed the following problem with which I have been struggling:
Prove that there exists a compact set $K \subset \mathbb{R}^n$ with empty interior and measure $0 \...
- 648
2
votes
1
answer
264
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Dominated convergence, and continuity of an integral function
Let $f:[0,1]\times [0,1]\to [0,1]$ be a measurable function with the properties:
$\color{red}{(1)}$ for every $x\in [0,1]$ is $y\mapsto f(x,y)$ continuous
$\color{blue}{(2)}$ for every $y\in [0,1]$ is ...
- 11.2k
4
votes
1
answer
237
views
How do I prove that this function is a measure?
I have the following problem:
Let $(\Omega, \mathfrak{A}, \mu)$ be a measurespace and let $f:\Omega \rightarrow [0,+\infty]$ be a nonegative simple function, i.e. $f(\Omega)=\{b_1,...b_s\}$. We ...
- 2,935
1
vote
1
answer
44
views
Solution verification: $m(A)>1$ implies $\exists x,y\in A$ s.t. $x-y\in\mathbb{N}.$
Let $A\subset\mathbb{R}$ with $m(A)>1$. Prove that there exists $x,y\in A$ such that $x-y$ is a positive integer. ($m$ is the Lebesgue measure.)
I think I have an interesting idea to solve this ...
- 285
0
votes
0
answers
114
views
Show that in Lebesgue theory $\int_\mathbb{R^2} f(\sqrt{x^2+y^2}) dxdy = 2 \pi \int_0^\infty r f(r) dr$
We want to show for any nonnegative integrable $f$ $\int_\mathbb{R^2} f(\sqrt{x^2+y^2}) dxdy = 2 \pi \int_0^\infty r f(r) dr$ (without Jacobian formula). This has probably been asked before but I want ...
- 73
1
vote
0
answers
49
views
Writing the iterated expectation with a single integral
I would like to write the expected value of $c(x)$ where $x$ is sampled from a distribution $\gamma(x|m)$ and $m$ is sampled from another distribution $\omega(m)$. Here, for any fixed $m$, the ...
0
votes
1
answer
194
views
Integral of product of characteristic functions
Let $(X, \mathcal{M},\mu), (Y,\mathcal{N},\nu)$ be measure spaces. Show for characteristic functions $\chi_A \in L^{+}(X),\chi_B \in L^{+}(Y)$, that $\chi_{A\times B} \in L^{+}(X \times Y)$ and $$\int\...
- 85