All Questions
Tagged with duality measure-theory
17 questions
4
votes
1
answer
165
views
Dual spaces of Banach-valued $L^{p}$-spaces
Let $(\Omega,\mathcal{F},\mu)$ be a measure space (say complete and $\sigma$-finite, for simplicity). Furthermore, let $(X,\Vert\cdot\Vert_{X})$ be an arbitrary Banach space. I denote by $(L^{p}(\...
- 1,429
4
votes
0
answers
497
views
Fixing the duality $L^\infty(X)= L^1(X)^*$ for Radon measure spaces
Consider the following fragment from Folland's book "A course in abstract harmonic analysis":
Let me denote the Borel subsets of $X$ by $\mathscr{B}(X)$. Folland claims that if $\mu$ is a ...
- 175
5
votes
1
answer
496
views
Unifying two definitions of $L^\infty$
Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon measure on $X$.
Definitions:
A subset $E\subseteq X$ is called locally Borel if $F \cap E$ is Borel for every Borel set $F\subseteq X$ ...
- 175
1
vote
0
answers
191
views
Dual of union of Reproducing Kernel Hilbert Spaces
I have a union of Reproducing Kernel Hilbert Spaces $\mathcal{B}$. I am interested in finding the dual of $\mathcal{B}$. Knowing what the dual is might help to write an alternate formulation for the ...
- 171
8
votes
0
answers
1k
views
Wasserstein distance and Monge-Kantorovich-Rubinstein duality
The definition of Wasserstein $p$-distance between two measures $\mu$ and $\nu$ on a Polish space $X$ is given by
$$
W_p(\mu, \nu)^p = \inf_{\gamma \in \Pi(\mu, \nu)} \int_{X \times X} d(x, y)^p\; d\...
- 1,407
4
votes
3
answers
2k
views
Duality of finite signed measures and bounded continuous functions
Let $E$ be a metric space, $C_b(E)$ denote the space of bounded continuous functions $E\to\mathbb R$ (equipped with the supremum norm), $\mathcal M(E)$ denote the space of finite signed measures on ...
- 167
10
votes
1
answer
1k
views
Dual space of continuous Banach-space-valued functions
Let $X$ be a Banach space and $K$ some compact Hausdorff space. I am interested in the dual space of the Banach space
$$C(K; X) = \lbrace f: K \to X, \ f \text{ is continuous}\rbrace, \qquad \lVert ...
- 381
2
votes
3
answers
928
views
Is unit ball in 2-Wassestein metric weakly compact?
This might be a trivial question, but I am trying to prove equi-coerciveness of some family of functions on the space of Probability measures on some space. I could reduce the problem to showing that $...
- 371
3
votes
0
answers
188
views
Is the category of abstract $\sigma$-algebras the pro-category of countable discrete spaces?
I am wondering if the category of abstract $\sigma$-algebras is contravariantly equivalent to the pro-category of the category of countable sets.
Note: I have tweaked this question a little. Now it ...
user30211
9
votes
0
answers
217
views
Point-free topology, but with $\sigma$-algebras instead of spaces
I have a question about $\sigma$-algebras in relation to point-free topology. The question was inspired by a comment on a similar question I had:
If abstract $\sigma$-algebras (i.e. certain boolean ...
user30211
2
votes
0
answers
158
views
The dual of the space of continuous sections in a vector bundle
If $X$ is a compact Hausdorff space, one may view the space of complex, continuous functions on it as the space of continuous sections in the trivial Hermitian bundle $X \times \mathbb C$. By the ...
- 5,407
1
vote
0
answers
154
views
Weak$^*$ topology on Bochner $L^p$-spaces
Assume that $X$ is a Banach space such that the dual $X'$ has the Radon-Nikodym property. Moreover let $(\Omega,\Sigma,\mu)$ a say finite measure space. Then we know that for $1\leq p<\infty$ holds ...
2
votes
1
answer
702
views
Duality of Bochner $L^{\infty}$ space
Let's have a look to the unit interval $[0,1]$ and a Banach space $X$ and then to the space
$$
E:=L^{\infty}([0,1],X),
$$
i.e. all essentially bounded Banach-valued functions $f:[0,1]\rightarrow X$. ...
3
votes
0
answers
210
views
On Radon-Nikodym property of a dual of a Bochner space
Let ${\rm S}^{n-1}$ be the unit sphere of ${\bf R}^n$ and let us consider the dual of the space $L^p\left({\bf R}^n; C^1({\rm S}^{n-1})\right)$, for some $p\in\langle1,\infty\rangle$: it is the space ...
- 165
5
votes
1
answer
611
views
The topological duals of spaces of finite measures
In volume 1 of "Linear Operators", Dunford and Schwartz say that (footnote F1, page 374)
"No completely satisfactory representation for the conjugate space of $ba(S, \Sigma)$, $ca(S, \Sigma)$ or $...
- 5,407
2
votes
3
answers
3k
views
dual space of a subspace of the space of bounded measures
Let $\mathcal{M}=\mathcal{M}(\mathbb{R})$ be the space of bounded measures. Equipped with the weak convergence, the dual space of $\mathcal{M}$ is $\mathcal{C}_b(\mathbb{R})$ consisting of continuous ...
- 1,835
6
votes
4
answers
1k
views
Existence of dominating measure for weak*-compact set of measures
I have posted the following question also here a longer time ago, but due to no answers I thought it might fit better to MO.
Let $(\Omega,\mathcal F)$ be a measurable space and $\mathcal P$ a weak*-...
- 215