All Questions
Tagged with integration measure-theory
2,732
questions
3
votes
1
answer
68
views
Radon Nikodym derivative and distribution function
Let $\mathfrak{B}$ be the Borel $\sigma$-algebra over $\mathbb{R}$ and $\beta$ the Borel-Lebesgue measure over $\mathfrak{B}$. Let $\mu$ be a Borel measure over $\mathbb{R}$ s.t. the distribution ...
- 115
4
votes
1
answer
61
views
Finding a function which is $L^1$ not $L^2$ and the integral is bounded by the square root.
So I have been trying to solve the following this past exam problem:
Find $f\in L^1(\mathbb{R})$, not $L^2(\mathbb{R})$ with the property:
$$
\int_{A}|f(x)|dm\leq \sqrt{m(A)}\quad\text{ for all } A\...
- 394
2
votes
1
answer
54
views
$f_n \xrightarrow{d} f$ if and only if $f_n \xrightarrow{m} f$ in $(L^0([0,1]), d)$ where $d(f,g) = \int_0^1 \frac{|f(x)-g(x)|}{1 + |f(x)-g(x)|}dx$
Let $L^0([0,1])$ be the vector space of Lebesgue-measurable functions on $[0,1]$. Let $d$ be the metric on $L^0([0,1])$ given by $$d(f,g) = \int_0^1 \frac{|f(x)-g(x)|}{1 + |f(x)-g(x)|}\, dx.$$
Prove ...
- 14.2k
1
vote
1
answer
45
views
Weak convergence in $L^2$ equivalence
Problem statement: Denoting by $B_r$ the open ball of $\mathbb{R}^N$ centered at the origin with radius $r$, consider a sequence $f_n \in L^2(B_1)$ which is bounded in the $L^2$ norm. Prove that $f_n$ ...
- 608
3
votes
1
answer
109
views
Stokes theorem for currents on manifolds with corners
Let $M\subset\mathbb R^N$ be a compact oriented $n$-(sub)manifold with corners and $\omega$ be an $(n-1)$-form on it. The usual statement of Stokes theorem
$$\int_M d\omega=\int_{\partial M}\omega$$
...
- 1,853
1
vote
0
answers
29
views
About the behaviour of an integral for $|x| > 1$ and $|x| < 1$
Let $f = \chi_{B(0,1)}$. Can anyone help me with the behavior of the following convolution $$f * |\cdot|^{-\alpha}(x) = \int_{B(0,1)}\frac{1}{|x-y|^{\alpha}}dy,$$for the cases $|x| > 1$ and $|x| &...
- 796
0
votes
1
answer
64
views
$S = \{z^2 = x^2 + y^2 + 1, z > 0 \}$. Calculate $\int \int_S \frac{d \lambda_2}{z^4}$
$S = \{ z^2 = x^2 + y^2 + 1, z > 0 \}$. Calculate $\int \int_S \frac{d \lambda_2}{z^4}$.
So, first, I would change that into cylindrical coordinates, to get:
$x = rcos(\alpha)$
$y = rsin(\alpha)$
...
- 1,086
0
votes
1
answer
77
views
Show integral identity
Given a measure space $(\Omega,A,\mu)$ and a non-negative measurable function $f:\Omega \to \mathbb{R}$, show that
$\int\ f d\mu = \int_{[0,\infty)} \mu(\{f>x\}) d\lambda(x)$.
So I think you show ...
- 48
3
votes
2
answers
89
views
Question on Complex Integral with Polar Form
Let $f$ be a complex-valued integrable function. Write the complex number $\int fd\mu$ in its polar form, letting $w$ be a complex number of absolute value 1 such that
\begin{align}
\int fd\mu = w\...
- 2,493
4
votes
1
answer
96
views
Question about Proof of the Integrability of $f$ and $f_1,f_2,\dots,$ in Lebesgue's Dominated Convergence Theorem
I am self-studying measure theory and got stuck on part of the proof of the Lebesgue's Dominated Convergence Theorem:
Theorem$\quad$ 2.4.5$\quad$ (Lebesgue's Dominated Convergence Theorem) Let $(X,\...
- 2,493
3
votes
0
answers
50
views
Proof of Beppo Levi's Theorem [closed]
I am self-studying measure theory using Measure Theory by Donald Cohn. The text presented the following result but lack of detailed proof. I tried to write up the proof, and I would really appreciate ...
- 2,493
1
vote
1
answer
66
views
A problem about zero-measure set in manifold.
Let $M$ be an $n$-dimensional differentiable manifold. A subset $N \subset M$ is said to have zero measure if the sets $\varphi_\alpha^{-1}(N) \subset U_\alpha$ have zero measure for every ...
- 97
0
votes
1
answer
34
views
Tailsum Formula and Indicator Functions
In my probability theory class we proved that $$\mathbb{E}[x]=\int_0^\infty \mathbb{P}(X>t) dt,$$ where $X\geq0$ is a non-negative random variable and $\mathbb{E}[X]:= \int_\Omega X(\omega) d\...
- 325
0
votes
1
answer
45
views
If $f^{-1}(I)$ is a Borel set for every interval $I$, why is $f^{-1}(B)$ a Borel set for every Borel set $B$?
My book defines a Borel subset of an interval $X$ of $\mathbb R$ to be any set which is in every $\sigma$-field containing all finite unions of intervals in $X$. Then they define a function $f:X\to [0,...
- 411
3
votes
2
answers
53
views
If $\int_Afd\mu\geq0$ for all $A\in\mathscr{A}$, then $\int f\chi_Ad\mu=0$ for $A=\{x\in X:f(x)<0\}$
I am self-studying measure theory using Measure Theory by Donald Cohn. I am confused by his proof of the following result:
Corollary 2.3.13$\quad$ Let $(X,\mathscr{A},\mu)$ be a measure space, and ...
- 2,493