All Questions
Tagged with measure-theory integration
2,732
questions
2
votes
1
answer
41
views
A question about switching the order of $\int$ and $\lim$ for a series of complex functions
When I took undergraduate complex analysis, the instructor was trying to prove an inequality, and he used a technique as below:
$$\int \lim_{n\rightarrow \infty} f_n dz= \lim_{n\rightarrow \infty} \...
- 1,131
0
votes
0
answers
11
views
Integral of a function with respect to a measure on a mixed space $[\![ N ]\!]\times \mathsf{X}$
Task
$\newcommand{\intbrackets}[1]{[\![ #1 ]\!]}$
$\newcommand{\Pcal}{\mathcal{P}}$
I would like to better understand integration of a function $f$ with respect to a probability measure $\mu$ on a ...
- 5,247
2
votes
0
answers
39
views
Isometry between the dual space $L^1(\mu)^*$ and $L^\infty(\mu)$. In need of a 'trial function'.
I am currently trying to show that $L^p(\mu)^*$ is isometrically isomorphic to $L^{p'}(\mu)$, where p' is the dual exponent. We are working in a $\sigma$-finite measure-space $(X,\mathcal{E}, \mu)$.
I ...
1
vote
1
answer
95
views
How can I prove that $\Vert \int _Xfd\mu \Vert \leq \int _X\Vert f\Vert d\mu $?
Let $(X,\Sigma,\mu )$ be measure space. Denote by $\mathbb{R}^{m\times n}$ the set of the matrix $m\times n$. Suppose that $f:X\to \mathbb{R}^{m\times n}$ is a function whose coordinates are ...
- 1,209
0
votes
1
answer
46
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Prove $(x,y)\rightarrow f(x,y)=\dfrac {\cos(xy)}{(1+x^2)(1+y)}\in L^{1}(]0,+\infty×]-1,1[)$
today I have a question about prove this function
$$(x,y)\rightarrow f(x,y)=\dfrac {\cos(xy)}{(1+x^2)(1+y)}$$
Such that $(x,y)\in ]0,+\infty ×]-1,1[$ are integrable.
**My attempts **
I was use this ...
- 2,323
0
votes
1
answer
16
views
Measure is real if integration of all real valued continuous function is real
Let $X$ be a compact Harsdorff space and $\cal A$ be a $\sigma$-algebra over $X$. Let $\mu$ be a complex measure on $(X,\cal A)$. Let $C(X)$ denotes the set of all complex valued continuous function ...
- 2,007
3
votes
0
answers
87
views
Two formulations of Riesz–Markov–Kakutani representation theorem
Let $ X $ be a locally compact Hausdorff space, and $ C_c(X) $ be the space of all complex-valued continuous functions with compact support on $ X $.
As far as I know, there are two formulations of ...
- 884
0
votes
1
answer
34
views
Measure Theory: Proof regarding a measurability of a function [closed]
Is my proof for the following problem correct?
Let $f:X\rightarrow[0,\infty)$ and $X=\bigcup_{i\in\mathbb{N}}A_i$ be measurable.
Prove: $f$ is measurable $\iff$ $f\cdot\chi_{A_i}$ is measurable.
Proof:...
user1197316
6
votes
0
answers
126
views
How would one integrate over $SO(n)$?
Suppose you want to find the average of an $n\times n$ diagonal matrix $A$ over all possible rotations,
$$
\langle A\rangle = \int\limits_\text{SO($n$)} Q^T A Q \; dQ.
$$
It's easy enough to do this ...
- 61
0
votes
0
answers
25
views
Parameter Dependence Integrals
Let $(X,\mathcal{X},\lambda)$ be a measure space and $(\Theta,d)$ a metric space.
Assume a function $f:X\times\Theta\to \mathbb{R}$ be uniformly bounded, meaning $\sup_{x\in X,\theta\in\Theta}|f(x,\...
- 33
0
votes
1
answer
48
views
Nice application of dominated convergence theorem
Let $\delta \in \mathbb{R}$, $$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$$ for each $a>0.$
I am unable to find ...
- 645
1
vote
0
answers
92
views
On the mathematics behind the Dyson Series
I've come across the Dyson Series solution of the Schrödinger Equation arising in the interaction picture when dealing with a time dependent Hamiltonian. Since then I've been looking for a rigorous ...
0
votes
1
answer
104
views
Example of a function on $([0,1]; \sigma ([0;1]); \lambda )$ for which it has no meaning to write $ \int f d \lambda$
I am trying to understand Lebesgue integration and in order to understand well this concept I would like to have an example of a function $(0,1] $ ( or $ [0,1) , [0,1] , (0,1) $ ) for which it has no ...
- 719
0
votes
0
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62
views
Question concerning the correctness of this version of Fatou's Lemma
In lecture we learned about Fatou's Lemma stated as follows:
Let $(X, \mathcal{S}, \mu)$ be a measure space and $(f_k : X \to [0,\infty])$ measurable and $f: X \to [0, \infty] $ a function such that:
$...
- 615
0
votes
0
answers
43
views
Is it possible to define $L^p$ spaces using a non-sigma-finite measure space and a Banach space?
Most often (at least in probability), one defines the $L^p$ space as
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $p\geq 1$ be a real number. Then
$$
L^p(\Omega, \mathcal{F},...
- 5,247