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

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

2011-09-22

arxiv.org/abs/1109.4858v1

Mathematics

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Scientific paper

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.

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