Each Random Variable in separable Banach Space belongs to the Domain of Definition of some inverse to compact linear non-random operator

TL;DR

This paper proves that any random variable in a separable Banach space almost surely lies in the pre-image of some linear compact operator, providing a simple proof of this property.

Contribution

It introduces a straightforward proof that all random variables with Borelian distribution in separable Banach spaces are almost surely in the domain of some inverse of a compact linear operator.

Findings

Random variables in separable Banach spaces belong to the domain of some inverse of a compact operator with probability one.

The proof simplifies understanding the structure of random variables in Banach spaces.

Establishes a link between random variables and compact operators in functional analysis.

Abstract

We give in this short report a very simple proof that arbitrary random variable with Borelian distribution in separable Banach space belongs with probability one to a pre-image of some linear compact non-random operator.