Theorem of Galvin, Mycielski and Solovay

Question

I don't know is this the right place to ask this question, but can someone tell me where I can find the proof of the theorem of Galvin, Mycielski and Solovay. Theorem that says that a subset $X$ of real line has strong measure zero if, and only if, for each first category set $M$, the set $X+M$ is not the entire real line.

Answer 1

The Galvin–Mycielski–Solovay theorem is Theorem 3.5 of "Special subsets of the real line" by Arnold W. Miller, which is Chapter 5 of the Handbook of Set-Theoretic Topology; the proof is on pp. 209–210.

P.S. As suggested in a comment by Dave L. Renfro, that theorem can also be found in the book Set Theory: On the Structure of the Real Line by Tomek Bartoszyński and Haim Judah; it's Theorem 8.1.16 on p. 405, and the proof is on pp 405–406.

P.P.S. Another proof can be found in the paper:

Fred Galvin, Jan Mycielski, and Robert M. Solovay, "Strong measure zero and infinite games", Arch. Math. Logic 56 (2017), 725–732.

A version of that paper is available for free on Arnold W. Miller's website.