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Probability measure annihilating all finite-dimensional subspaces

Nizar El Idrissi, Hicham Zoubeir

TL;DR

The paper addresses the problem of constructing a probability measure on an infinite-dimensional Banach space $E$ that annihilates all finite-dimensional subspaces. It introduces an explicit, constructive method: embed $(0,1)$ into $E$ via an injective measurable map $x:(0,1)\to E$ built from a number-theoretic selection of index sets $N_t$ and a biorthogonal system, and define the measure $P$ as the pushforward of Lebesgue measure, $P(M)=\lambda(x^{-1}(M))$. The core contribution is proving that $P(M)=0$ for every finite-dimensional subspace $M\subset E$ because $M\cap \operatorname{Range}(x)$ is finite, leveraging the atomless nature of $\lambda$ and the injectivity of $x$. This establishes the existence of an atomless measure on $E$ with a strong annihilation property and addresses measurability of $\operatorname{Range}(x)$ as part of the construction.

Abstract

We construct a probability measure annihilating all finite-dimensional subspaces on an arbitrary infinite-dimensional Banach space.

Probability measure annihilating all finite-dimensional subspaces

TL;DR

The paper addresses the problem of constructing a probability measure on an infinite-dimensional Banach space that annihilates all finite-dimensional subspaces. It introduces an explicit, constructive method: embed into via an injective measurable map built from a number-theoretic selection of index sets and a biorthogonal system, and define the measure as the pushforward of Lebesgue measure, . The core contribution is proving that for every finite-dimensional subspace because is finite, leveraging the atomless nature of and the injectivity of . This establishes the existence of an atomless measure on with a strong annihilation property and addresses measurability of as part of the construction.

Abstract

We construct a probability measure annihilating all finite-dimensional subspaces on an arbitrary infinite-dimensional Banach space.
Paper Structure (3 sections, 2 theorems, 18 equations)

This paper contains 3 sections, 2 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Proof of the theorem
  3. Measurability of Range(x)

Key Result

Theorem 1.1

Let $(\Omega, \mathfrak{F}, P)$ be a probability space and $H$ be a real Hilbert space of infinite dimension. Let $(e_n)_{n \in \mathbb{N}}$ be a sequence of random variables defined on $(\Omega, \mathfrak{F}, P)$ and with values in $H$. Let $Q$ be a complete $\sigma$-finite probability measure on $

Theorems & Definitions (2)

  • Theorem 1.1
  • Theorem 1.2