# A lower density operator for the Borel algebra

## TL;DR

This paper establishes that the existence of a Borel lower density operator for countable sets in an uncountable Polish space is equivalent to the Continuum Hypothesis, linking set-theoretic axioms with measure-theoretic properties.

## Contribution

It proves the equivalence between the existence of a Borel lower density operator and the Continuum Hypothesis in uncountable Polish spaces.

## Key findings

- Existence of Borel lower density operator is equivalent to CH.
- Provides a characterization linking measure theory and set theory.
- Highlights the dependence of certain measure-theoretic constructs on set-theoretic axioms.

## Abstract

We prove that the existence of a Borel lower density operator (a Borel lifting) with respect to the $\sigma$-ideal of countable sets, for an uncountable Polish space, is equivalent to the Continuum Hypothesis.