Probability measure annihilating all finite-dimensional subspaces

TL;DR

This paper introduces a prime numbers-based method for constructing probability measures on infinite-dimensional Banach spaces that annihilate all finite-dimensional subspaces, expanding the toolkit beyond Gaussian and product measures.

Contribution

It presents a novel prime numbers-based construction method for such measures, demonstrating their genericity and establishing the existence of a large family of independent vectors.

Findings

Probability measures annihilating all finite-dimensional subspaces are generic.

Existence of an uncountable family of independent vectors in any infinite-dimensional Banach space.

New construction method supplements existing Gaussian and product measures.

Abstract

We propose in this short note a prime numbers-based method for constructing probability measures on infinite-dimensional Banach spaces annihilating all finite-dimensional subspaces, supplementing the methods of construction of Gaussian measures and infinite-product-type probability measures. This new method confirms that probability measures with this property are generic amongst probability measures that are not supported on finite-dimensional subspaces. In the process, we show the existence of an uncountable measurable family of independent vectors having the cardinality of the continuum in any infinite-dimensional Banach space.