I meant to write a longer blog entry on strong measure zero sets (on the real line ), but it is getting too long, so it may take me more than I expected. For now, let me record here an argument showing the following:
Theorem. If is a strong measure zero set and is a closed measure zero set, then has measure zero.
The argument is similar to the one in
Janusz Pawlikowski. A characterization of strong measure zero sets, Israel J. Math., 93 (1), (1996), 171-183. MR1380640 (97f:28003),
where the result is shown for strong measure zero subsets of . This is actually the easy direction of Pawlikowski’s result, showing that this condition actually characterizes strong measure zero sets, that is, if is measure zero for all closed measure zero sets , then is strong measure zero. (Since this was intended for my analysis course, and I do not see how to prove Pawlikowski’s argument without some appeal to results in measure theory, I am only including here the easy direction.) Pawlikowski’s argument actually generalizes an earlier key result of Galvin, Mycielski, and Solovay, who proved that a set has strong measure zero iff it can be made disjoint from any given meager set by translation, that is, iff for any meager there is a real with disjoint from .
I proceed with the (short) proof after the fold.