Skip to main content

Questions tagged [vector-measure]

Vector(-valued) measures are (finitely / countably) additive mappings from a sigma algebra to a Banach space, to which one can associate a scalar measure, it total variation measure.

Filter by
Sorted by
Tagged with
2 votes
0 answers
34 views

Density of finite-dim vector-valued measures

I'm trying to understand contour integrals $I(f, \Gamma):=\int_\Gamma f(\gamma)d\gamma$ in complex analysis and functional analysis as rigorously as possible. It seems to me this has natural ...
user760's user avatar
  • 1,424
0 votes
1 answer
34 views

Relation between the distance between two complex measures and their total variations.

Consider measure space $(X,\mathcal{A})$ and space of complex measures on $X$, denoted by $\mathcal{M}_{C}(X)$. For $\mu \in \mathcal{M}_C(X)$, we define total variation of $\mu$, as $\vert \mu \vert(...
MI00's user avatar
  • 277
1 vote
0 answers
87 views

vector measure finite variation

Let $(X, \mathcal{A})$ be a measurable space and $B$ a Banach space. Let $\mu: \mathcal{A} \to B$ be a countably additive vector measure. For a measurable function $f:X \to \mathbb{R}$. I want to ...
yf297's user avatar
  • 79
5 votes
1 answer
176 views

Definition of a finite vector-valued Radon measure: isn't this condition vacuous?

In Diestel & Uhl's "Vector measures" one finds the following definition 1 on page 1. Definition 1 (Countably additive vector measure). Let $X$ be a Banach space an $\mathcal F$ a $\sigma$...
ViktorStein's user avatar
  • 4,838
2 votes
1 answer
342 views

The C$^*$ algebras of matrices, continuous functions, measures and matrix-valued measures / continuous functions and their state spaces

I have no background in Algebra, but want to understand matrix-valued measures and matrix-valued continuous functions from the C$^*$-algebra perspective to identify what definition of matrix-valued ...
ViktorStein's user avatar
  • 4,838
1 vote
0 answers
87 views

Functional derivative of $u \mapsto R\left(\int_{\Theta} \phi(\theta) \; \text{d}u(\theta)\right) + \lambda u(\Theta)$ for matrix-valued measure $u$

Let $\Theta$ be a closed connected manifold, $H$ and $F$ Hilbert spaces and $R \colon H \to \mathbb R$ and $\phi \colon \Theta \to F$ smooth. What is the functional derivative of $$f(u) := R\left(\...
ViktorStein's user avatar
  • 4,838
1 vote
1 answer
186 views

What does this notation (inner product with differential) mean?

The following symbols are used in On optimal transport of matrix-valued measures by Yann Brenier and Dmitry Vorotnikov (cf. p. 3): $A : B := \text{tr}(A B^{\mathsf{T}}) = \sum_{i, j = 1}^{d} a_{i, j} ...
ViktorStein's user avatar
  • 4,838
2 votes
0 answers
162 views

Explicit form for the total variation measure of a symmetric matrix-valued measure

Let $X$ be a locally compact Polish space. Consider the set $M(X; S^d)$ of real symmetric matrix-valued measures on $X$, that is, the set of countably (with respect to the Frobenius norm $\| \cdot \|...
ViktorStein's user avatar
  • 4,838
3 votes
1 answer
172 views

Riesz–Markov–Kakutani representation for symmetric matrix-valued measures

Let $\Theta$ be a closed $d$-manifold (and hence a metric space with Borel-$\sigma$-algebra $\Sigma \subset 2^\Theta$), $S^d$ the set of real symmetric $d \times d$ matrices, which we can identify ...
ViktorStein's user avatar
  • 4,838