A subset of ℝ is meagre if it is a countable union of nowhere dense subsets (a set is nowhere dense if every open interval contains an open subinterval that misses the set).

Any countable set is meagre. The Cantor set is nowhere dense, so it's meagre. A countable union of meagre sets is meagre (e.g. all rational translates of the Cantor set).

There can also be meagre sets of positive measure, like "fat Cantor sets". To form a fat Cantor set, you start with a closed interval, then remove some open interval from the middle of it, then remove some open intervals from the remaining intervals, and so on. The result is nowhere dense because you removed open intervals all over the place. If the sizes of the intervals you remove get small fast, then the result has positive measure.

So does meagreness have any connection at all to measure? Specifically, are all measure zero sets meagre?

meagerand UK spellingmeagre. I suppose this goes along withtheater/theatreand similar. $\endgroup$ – Gerald Edgar Oct 25 '10 at 15:16meager. $\endgroup$ – Anton Geraschenko Oct 25 '10 at 17:56neighbourhoodso other UK spellings are not unexpected. But shouldn't it then be Wikipaedia? $\endgroup$ – Gerald Edgar Oct 25 '10 at 23:31