Cohen real or random real: effect on strong measure zero sets and strongly meager sets

TL;DR

This paper investigates how adding Cohen, random, or Hechler reals affects the measure and category properties of ground-model reals, showing they become strongly meager or have strong measure zero after such extensions.

Contribution

It demonstrates that adding a single Cohen, random, or Hechler real makes the ground-model reals strongly meager or of strong measure zero, revealing new interactions between forcing and measure/category properties.

Findings

Ground-model reals become strongly meager after adding a Cohen real.

Ground-model reals have strong measure zero after adding a Hechler real.

Adding a random real also makes ground-model reals strongly meager.

Abstract

We show that the set of the ground-model reals has strong measure zero (is strongly meager) after adding a single Cohen real (random real). As consequence we prove that the set of the ground-model reals has strong measure zero after adding a single Hechler real.