All Questions
Tagged with integration measure-theory
2,732
questions
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
5
votes
0
answers
136
views
Property of vector-valued measure
Let $B$ be a Banach space, let $(X,\mathcal{A})$ be a measurable space, and let $\mu:\mathcal{A}\to B$ be a vector-valued measure of bounded variation.
In general, if $B$ doesn't have the Radon-...
- 8,127
0
votes
0
answers
67
views
Some questions about integration and operator theory.
As we all know there are multiple integral operators which all basically do the same thing in various contexts. I am talking about operators like the Lebegues integral, Riemann integral and more ...
- 103
0
votes
0
answers
20
views
Integration over measure which counts jump discontinuities of specific length.
$f:[0,1] \to [0,1]$ is a monotonically increasing function with $f(0)=0$ and $f(1)=1$. Let $p>1$.
Define, for $0\leq a < b \leq 1$, $\mu((a,b])$ as the number of points $x \in [0,1]$ such that
$\...
2
votes
0
answers
66
views
Question on the Construction of the Integral in Measure Theory
I am self-studying measure theory, and I got some trouble understanding the construction of the integral. Here is the first two stages of the construction:
Stage 1$\quad$ We begin with the simple ...
- 2,493
2
votes
1
answer
42
views
Question About Proof of Proposition 2.3.1 in Measure Theory by Donald Cohn
I am self-studying Donald Cohn's Measure Theory, and I got stuck on a step in his proof of Proposition 2.3.1. Here is the statement of the proposition as well as its proof until the point where I have ...
- 2,493
0
votes
1
answer
16
views
Equality of probability measures from given integral
Let $(X,\mathcal{A})$ be a measurable space. Consider $\Bbb{R}$ equipped with its Borel sigma-algebra, and let $\mu,\nu$ be probability measures on $X\times\Bbb{R}$, with the product sigma-algebra.
...
- 8,127
0
votes
0
answers
33
views
Use of dominated convergence theorem in Manski (1985)
I'm confused by the use of the dominated convergence theorem (DCT) in Lemma 5 of Manski (1985) (see below). Note that $b = (\tilde{b}_1, \dots, \tilde{b}_{K-1}, b_K).$
Specifically:
I presume the ...
- 147
0
votes
1
answer
26
views
Construction of a simple function by open sets
A simple function $\phi: \mathbb{R} \to \mathbb{R}$ can be written as:
$$\phi(x) = \sum_{k=1}^n a_k \chi_{E_k}(x)$$
where each $E_k$ is a measurable set and $\cup E_k = \mathbb{R}$ and $\chi_{E_k}$ is ...
- 751
0
votes
0
answers
21
views
Antidivergence of function is Lipschitz
I was reading this paper https://epubs.siam.org/doi/pdf/10.1137/11081986X and they claim (page 672 and 673) that the flow field $v$ defined as the antidivergence of the density $\rho$ is Lipschitz ...
- 483
0
votes
0
answers
100
views
Convenience formula for integral on product space
Let $\Omega_1 \times \Omega_2$ equipped with some product $\sigma$-algebra be a product space.
Suppose $\mu$ is any positive measure (not necessarily any product measure) on $\Omega_1 \times \Omega_2$ ...
- 2,497
0
votes
1
answer
59
views
Detail in standard measure theory I cannot seem to obtain
There is a standard result in measure/integration theory which I just cannot seem to obtain.
If $f \colon X \to \mathbb{C}$ is measurable ($X$ is any measurable space), there exist simple measurable ...
- 338
1
vote
0
answers
22
views
If $C_1$ and $C_2$ are uniformly integrable then $\{f_1+f_2:f_1\in C_1, f_2\in C_2\}$ is?
Let $C_1$ and $C_2$ be uniformly integrable collections of functions
on a measure space $(\Omega,\mathscr{F},\mu)$,
and let $C$ be the collection defined as
\begin{equation*}
C = \{f_1 + f_2:f_1\in ...
- 1,148
1
vote
0
answers
39
views
Is every probability mass function $f_X$ of a random variable $X$ the Radon-Nikodym derivative of $X_*P$ with respect to the counting measure?
Let $X$ be a discrete random variable (meaning $\text{im}(X)$ is countable) from a probability triple $(A,\mathcal{A},P)$ to a measurable space $(\mathbb{R},\mathcal{B})$ where $\mathcal{B}$ is the ...
- 5,166
0
votes
0
answers
18
views
Properties of real representation of integral over complex weight function
Consider some positive and real-valued functions $p, \tilde p :\mathbb{R}^2\to [0,\infty)$ that are integrable and decay sufficiently fast at infinity. For any entire function $f: \mathbb{C} \to \...
- 1,134