Would a conserved Lebesgue measure have been acceptable?
Not unless you enjoy anachronisms. The Greeks probably wouldn't have liked that either; the basic point stands: if at any point one can divide it to produce a void and 2 smaller masses, then where is the 'real' mass? Any point you pick, I can turn into void. If I can do it for any point, I can do it for every point, and if every point is void...
Unless you postulate a knife with really weird properties, cutting a continuous object in half isn't turning matter into void. It's moving some of the matter without changing its density (hence my offer of conservation of volume). You can do that to every point currently occupied by an object, but only by reserving an equal amount of space that's currently void and displacing all of the matter to there.
This is our monthly thread for collecting these little gems and pearls of wisdom, rationality-related quotes you've seen recently, or had stored in your quotesfile for ages, and which might be handy to link to in one of our discussions.