@George: Thanks for your patience. My first question is that, in Step 2, you write B as a union of countably many open balls of radius at least r, but how can we control the radius of the balls in the countable union (to make it $\geq r$)? My second question is answered by the details you added. Thanks!

George, I still have two questions concerning your sketch of proof. First, how can you guanrantee each of the open balls in the countable union has radius greater than or equal to 1? Second, I don't know how to use convexity to prove $\mu (B') \leq (1+\epsilon)^{N}\mu(B)$

We are considering non-trival closed balls. So a single point is not a closed ball.