All Questions
327
questions
1
vote
0
answers
68
views
What's the definition of a line integral on a possibly disconnected curve?
I'm trying to understand this paper, and I see this integral (page 2):
$$
\int_{B\ \cap\ \mathcal{C}} (1 - y)dy,
$$
where $\mathcal{C}$ is the curve given by $x = ye^{1-y}$ and $B$ can be any ...
- 1,961
0
votes
0
answers
62
views
Proof that the volume function is $\sigma$-additive
A $\textbf{box}$ $Q\subset \mathbb{R}^n$ is defined as $Q:= (a_1,b_1) \times \ldots \times (a_n,b_n)$ and the same with closed or half opened intervals where $a_i < b_i \in \mathbb{R}$ and in case ...
- 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
0
votes
1
answer
48
views
Nice application of dominated convergence theorem
Let $\delta \in \mathbb{R}$, $$f(x)=\frac{sin(x^2)}{x}+\frac{\delta x}{1+x}.$$
Show that $$\operatorname{lim_{n\to \infty}} \int_{0}^{a}f(nx)=a\delta$$ for each $a>0.$
I am unable to find ...
- 645
0
votes
1
answer
52
views
Example of sequence of integrable maps such that integral smaller than $1/n$ but doesn't converge to zero almost everywhere
I am looking for a measure space $(\Omega,\mathcal{M},\mu)$ and a sequence of integrable functions $(f_n)$ in $\mathcal{L}^1(\Omega,\mathcal{M},\mu)$ with the property that
$\int_\Omega |f_n|d\mu \leq ...
- 415
0
votes
1
answer
147
views
Prove $\nu$ is absolutely continuous to Lebesgue measure if and only if $f$ is absolutely continuous.
Let $\nu$ be a finite Borel measure on $[0,1]$. Define $f : [0,1] \to \mathbb R$ by $f(x) = \nu ([0,x))$. Prove $\nu$ is absolutely continuous to Lebesgue measure (\mu) if and only if $f$ is ...
- 1,575
1
vote
1
answer
58
views
Calculating a series with the Beppo-Levi Theorem
As an excercise of Measure Theory, I have to calculate for all $a\in [-1,1]$ that: $$\int_{(0,+\infty)} \frac{\sin (t)}{e^t - a} d\lambda(t) = \sum_{n\geq 1} \frac{a^{n-1}}{n^2+1} $$
where $\lambda$ ...
- 731
3
votes
0
answers
66
views
Calculating an integral depending on a parameter
I have to prove as an excercise of my Measure Theory lessons that for $p > -1$ it is true that: $$\int_0^1 \frac{-\ln(x) x^p}{1-x} dx = \sum_{n \geq 1} \frac{1}{(n+p)^2}$$
As an advice, the ...
- 731
0
votes
1
answer
94
views
On the layer cake representation
The statement, as in Lieb-Loss Analysis, for the Layer cake representation is:
Let $\nu$ be Borel measure on $[0,\infty)$ such that $\phi(t) := \nu([0,t))$ is finite for each $t > 0$. Now let $(\...
- 796
0
votes
0
answers
29
views
Does this integral converge over a positive measure?
Let $\lambda$ be a positive measure on $(0;+\infty)$ satisfies
$\int_{(0,+\infty)} (1+s)^{-1} d \lambda (s) < +\infty$. Does this
integral converge?
$$\displaystyle\int_{(0;+\infty)} \dfrac{1}{s} d ...
- 983
0
votes
1
answer
98
views
Substitute measure of integration
Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X:\Omega \rightarrow \mathcal{X}$ and $Y:\Omega \rightarrow \mathcal{Y}$ random variables. Let $f:\mathcal{Y}\rightarrow \mathbb{R}$...
- 365
1
vote
0
answers
61
views
If $m(A)=0$, does this imply that $m(\log(A)) = 0$ when $A\subset \mathbb{R}_{+}$
The question is as follows:
Let $A\subset \mathbb{R}_{+}$ and let $\log(A)=\{\log(t): t\in A\}$.If $m(A)=0$,then is it true that $m(\log(A)) = 0$? If $m(A) < \infty$, then is it true that $m(A) <...
- 13
2
votes
1
answer
91
views
Why regularity is important for a Borel measure?
Why the property of being regular for a measure is important and what are some important and essential theorems that use this properties to hold true ?
Thanks in advance.
- 107
3
votes
3
answers
125
views
Convergence a.e. of $\sum_{n=1}^{\infty} \frac{a_n}{\sqrt{\left|x-r_n\right|}}$, where $\sum_{n=1}^{\infty}\left|a_n\right|<\infty$
Let $\left\{a_n\right\}_{n=1}^{\infty}$ and $\left\{r_n\right\}_{n=1}^{\infty}$ two sequences of real numbers, and suppose that: (i) $\sum_{n=1}^{\infty}\left|a_n\right|<\infty$; (ii) $r_i \neq r_j$...
0
votes
2
answers
55
views
Proving Lebesgue integrability of a bounded function
Let $u:\mathbb{R}\to\mathbb{R}$ be a simple function such that \begin{equation} |u(x)|\le\frac{x^2}{1+|x|^3}\end{equation} for all $x\in\mathbb{R}$. Prove that $u$ is Lebesgue integrable.
My attempt: ...
- 135