A covering theorem and the random-indestructibility of the density zero ideal

TL;DR

The paper proves a covering theorem in measure spaces and demonstrates that the ideal of density zero subsets of natural numbers remains unaffected by random forcing, showing its indestructibility.

Contribution

It introduces a new covering theorem involving density zero sequences and provides a simple proof of the density zero ideal's random-indestructibility.

Findings

Existence of a density-zero subsequence still covering almost every point infinitely often

Density zero ideal is shown to be random-indestructible

Probabilistic construction technique used in the proof

Abstract

The main goal of this note is to prove the following theorem. If $A_n$ is a sequence of measurable sets in a $\sigma$-finite measure space $(X, \mathcal{A}, \mu)$ that covers $\mu$-a.e. $x \in X$ infinitely many times, then there exists a sequence of integers $n_i$ of density zero so that $A_{n_i}$ still covers $\mu$-a.e. $x \in X$ infinitely many times. The proof is a probabilistic construction. As an application we give a simple direct proof of the known theorem that the ideal of density zero subsets of the natural numbers is random-indestructible, that is, random forcing does not add a co-infinite set of naturals that almost contains every ground model density zero set. This answers a question of B. Farkas.