All Questions
10 questions
9
votes
3
answers
3k
views
What is the Dunford Integral and why is it useful?
Wikipedia http://en.wikipedia.org/wiki/Pettis_integral defines the Pettis Integral for Banach space valued functions wrt to some measure space by duality.
It calls the Pettis & Bochner integral ...
- 2,083
6
votes
1
answer
223
views
Is the space of vectorial functions that are Dunford integrable complete?
Let $X$ a Banach Space and $(\Omega, \Sigma, \mu)$ a measure space. A function $F:\Omega\rightarrow X$ is Dunford integrable if $x^\ast\circ F$ is $\mu$-integrable for every $x^\ast\in X^\ast$. The ...
5
votes
1
answer
914
views
Extension of a function from almost everywhere to everywhere
The informal general question is: let $f$ be a "sufficiently nice" function, defined "almost everywhere". Can we develop a method to uniquely extend $f$ to the "remaining" points?
Example: Let $f(x)=\...
- 7,182
4
votes
1
answer
548
views
Two definitions of $L^p$ spaces that are not always equivalent
There are two definitions of $L^p(S, \Sigma,\mu)$ in the literature. (Here $S$ is a set, $\Sigma$ is a $\sigma$-algebra of subsets of $S$ and $\mu$ is a positive measure.) The two definitions are ...
- 41
4
votes
1
answer
819
views
The Notion of Strong Measurability for Separable Banach Spaces
Let $ (X,\Sigma,\mu) $ be a measure space and $ B $ a Banach space. According to my understanding, a function $ f: X \to B $ is said to be strongly $ \mu $-measurable if and only if it is the almost-...
- 1,396
3
votes
1
answer
197
views
Post composition of integral
Setup:
If $\langle \Omega, \mathfrak{F},\mu \rangle$ is a measure space, $f:\Omega \rightarrow E$ is a weakly-measurable function to a Banach space $E$, $g: E \rightarrow E'$ is a diffeomorphism and ...
- 5,405
3
votes
0
answers
278
views
Radon-Nikodym derivative of vector-valued measure with respect to another vector-valued measure
Let $(X, | \cdot |)$ be a Banach space.
I am interested in whether one can extend the definition of the Kullback-Leibler divergence
$$
\text{KL}(\mu \ \Vert \ \nu)
:= \int_{\Omega} \ln\left(\frac{\...
- 67
2
votes
0
answers
75
views
Have the open Questions 1 and 2 from Section 7 of the paper "Integrals with values in Banach spaces" been answered?
Some context: I had previously asked the post below on MSE, but someone suggested I ask it here and delete the original post.
In section 7 of the paper Integrals with values in Banach Spaces and ...
- 121
1
vote
3
answers
580
views
Squeezing more convergence from the convergence in all $L^p$ spaces
Let $X$ be a space endowed with a finite measure $m$. Let $f_n : \to \mathbb C$ be measurable functions such that $|f_n| \le 1$ for all $n$ and $f_n \to 0$ in every space $L^p (X)$ with $p \in [1, \...
- 5,407
1
vote
1
answer
89
views
Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace - Part II
This is a follow-up to this previous question, but under stronger assumptions.
Let $(X, \mu)$ be a (say, $\sigma$-finite) measure space, let $g \in L^2$ (say, over the real
scalar field). Let $\tilde ...
- 12.6k