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.
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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 $\sigma$...
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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 ...
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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(\...
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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} ...
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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 \|...
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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 ...
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