# 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.

6 questions
Filter by
Sorted by
Tagged with
121 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 $\sigma$...
• 4,436
92 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 ...
• 4,436
1 vote
74 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(\...
• 4,436
1 vote
153 views

The following symbols are used in On optimal transport of matrix-valued measures by Yann Brenier and Dmitry Vorotnikov (cf. p. 3): $A : B := \text{tr}(A B^{\mathsf{T}}) = \sum_{i, j = 1}^{d} a_{i, j} ... • 4,436 2 votes 0 answers 89 views ### Explicit form for the total variation measure of a symmetric matrix-valued measure Let$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 \|...
• 4,436
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 ...