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Control Measures for Bochner $L_{0}$-Valued Vector Measures

Lech Drewnowski, Alexandre Reggiolli Teixeira

Abstract

It is shown that for any finite positive measure $μ$ defined on a measure space $(S, Σ)$, and any Banach or Fréchet space $Z$, the control measure Theorem of Talagrand (T) is true for the case when the (stochastic) vector measure $\boldsymbol{m}:\mathcal{E} \to L_0(μ,Z)$, defined on another measurable space $(E, \mathcal{E})$, takes values in $L_{0}(μ,Z)$, the Bochner space of vector-valued functions associated to $μ$ and $Z$. As a consequence, we also obtain a Rybakov type result for this control. Finally, we give the relation of this result to bounded multiplier properties (BMP) of $F$-spaces and pose various open problems related to it.

Control Measures for Bochner $L_{0}$-Valued Vector Measures

Abstract

It is shown that for any finite positive measure defined on a measure space , and any Banach or Fréchet space , the control measure Theorem of Talagrand (T) is true for the case when the (stochastic) vector measure , defined on another measurable space , takes values in , the Bochner space of vector-valued functions associated to and . As a consequence, we also obtain a Rybakov type result for this control. Finally, we give the relation of this result to bounded multiplier properties (BMP) of -spaces and pose various open problems related to it.
Paper Structure (8 sections, 13 theorems, 89 equations)

This paper contains 8 sections, 13 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Basic notions and notation
  3. More notions, notation and conventions
  4. Preliminary facts
  5. Closed subspaces of $Z$ and their counterparts in $L_0(\mu,Z)$
  6. The existence of control measures for $L_0(\mu,Z)$-valued vector measures
  7. F-spaces with BMP and convexly bounded vector measures
  8. Some Elements of Submeasure Control for Non-Locally Convex Spaces

Key Result

Lemma 2.4

A function $g\in \mathcal{L}_0(\mu, Z)$ is $\mu$-equivalent to some function $f\in \mathcal{L}_0(\mu,X)$ iff $g(S)\overset{\mu}{\subset} X$.

Theorems & Definitions (30)

  • proof
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 3.2
  • Theorem 3.3: Controls for $L^{0}$ Böchner-Valued Vector Measures
  • proof
  • Remark 3.4
  • ...and 20 more