Let E be a measurable set
define d(x, E) to be the density of E at x, where
d(x, E) =
[COLOR=Black]where B is a closed ball centered (this is not the usual def'n of density, I know) at x. I.e. the balls are converging to x.
a) If x is in the boundary of E, is 0 < d < 1?
b) If 0 < d < 1, is x in the boundary of E?
Here are my answers, presented w/o proof.
a) No, look at any set of measure 0, then d = 0
b) Yes, cause every open ball will intersect both E and its compliment
Do these answers look like they might be wrong? Otherwise, I'll assume my proofs are correct