All Questions
243
questions
1
vote
1
answer
35
views
Prove: If $f_1$ and $f_2$ are integrable, then $f_1\vee f_2$ is integrable over each $A\in\mathscr{A}$.
I need to prove the following result:
Let $(X,\mathscr{A},\mu)$ be a measure space. If $f_1$ and $f_2$ are integrable, then $f_1\vee f_2$ is integrable over each $A\in\mathscr{A}$.
Here is my ...
- 2,493
1
vote
0
answers
58
views
Prove: $F:M(X,\mathscr{A},\mathbb{R})\to\mathbb{R}$ defined by $F(\mu)=\int fd\mu$ is a linear functional. [closed]
I need to prove the following result:
Remark 4.30$\quad$ Let $M(X,\mathscr{A},\mathbb{R})$ be the collection of all finite signed measures on $(X,\mathscr{A})$. Let $B(X,\mathscr{A},\mathbb{R})$ be ...
- 2,493
4
votes
1
answer
55
views
If $g_1, g_2\in\mathscr{L}^{\infty}(X,\mathscr{A},\mu)$ are equal locally $\mu$-almost everywhere, then $T_{g_1}=T_{g_2}$.
Background
Suppose that $(X,\mathscr{A},\mu)$ is an arbitrary measure space, that $p$ satisfies $1\leq p<+\infty$, and that $q$ is defined by $\frac{1}{p}+\frac{1}{q}=1$. Let $g$ belong to $\...
- 2,493
3
votes
1
answer
67
views
Prove that $\int _Xf_ngd\mu \overset{n\to\infty}{\to}\int _Xfgd\mu$ ,$\forall g\in \mathcal{L}^\infty (\mu )$ if it's true $\forall g\in C_b(X)$
Let $X$ be a Polish space and $\mu :\mathfrak{B}_X\to\overline{\mathbb{R}}$ a finite measure on the Borel subsets of $X$. Suppose $(f_n)_{n\in\mathbb{N}}$ is a sequence of $\mathcal{L}^1(\mu )$ and $f\...
- 1,209
14
votes
4
answers
3k
views
Why Learn Measure Theory and Lebesgue Integration?
As someone who has taken two semesters of real analysis, having been exposed to the rigorous definition of the Riemann-Stieltjes integral - why should I learn Lebesgue integration? The Riemann ...
- 365
3
votes
1
answer
51
views
$\mathscr{L}^{p_1}(\mathbb{N},\mathscr{A},\mu)\subseteq\mathscr{L}^{p_2}(\mathbb{N},\mathscr{A},\mu)$: $1\leq p_1<p_2<+\infty$, $\mu$ counting measure
I need to prove the following:
Suppose that $1\leq p_1<p_2<+\infty$. Let $\mu$ is the counting measure on the $\sigma$-algebra $\mathscr{A}$ of all subsets of $\mathbb{N}$. Then $\mathscr{L}^{...
- 2,493
0
votes
1
answer
55
views
"Leibniz's rule" for $t\in\mathbb{R}^n$
I am looking for a reference giving a measure-theoretic proof of a claim from the German Wikipedia. I have searched the references given on that site, as well as the English speaking Wikipedia and all ...
- 1
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
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
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
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
3
votes
1
answer
58
views
Question About Function Integrability - Proposition 2.3.10 from Measury Theory by Donald Cohn
I am self-studying measure theory using Measure Theory by Donald Cohn. The book makes the following definition:
Definition$\quad$ Suppose that $f:X\to[-\infty,+\infty]$ is $\mathscr{A}$-measurable ...
- 2,493