## Measure theory Questions

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