On strong measure zero sets

December 6, 2013

I meant to write a longer blog entry on strong measure zero sets (on the real line \mathbb R), 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 X is a strong measure zero set and F is a closed measure zero set, then X+F 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 \{0,1\}^{\mathbb N}. This is actually the easy direction of Pawlikowski’s result, showing that this condition actually characterizes strong measure zero sets, that is, if X+F is measure zero for all closed measure zero sets F, then X 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 X has strong measure zero iff it can be made disjoint from any given meager set by translation, that is, iff for any G meager there is a real r with X+r disjoint from G.

I proceed with the (short) proof after the fold.

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