Is an arbitrary union of non-trivial closed balls in the Euclidean space $\mathbb R^{N}$ Lebesgue measurable? If so, is it a Borel set?

@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 $\mu (B') \leq (1+\epsilon)^{N}\mu(B)$

Is an arbitrary union of non-trivial closed balls in the Euclidean space $\mathbb{R}^n$ Lebesgue measurable? If so, is it a Borel set?

@George

I still have two questions concerning your sketch of proof.

First, how can you guarantee 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 $\mu (B') \leq (1+\epsilon)^{N}\mu(B)$

Is an arbitrary union of non-trivial closed balls in the Euclidean space $\mathbb R^{N}$ Lebesgue measurable? If so, is it a Borel set?

Is an arbitrary union of non-trivial closed balls in the Euclidean space $\mathbb R^{N}$ Lebesgue measurable? If so, is it a Borel set?

@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 $\mu (B') \leq (1+\epsilon)^{N}\mu(B)$

Is an arbitrary union of non-trivial closed balls in the Euclidean space R^{N} Lebesgue measurable? If so, is it a Borel set?

Is an arbitrary union of non-trivial closed balls in the Euclidean space $\mathbb R^{N}$ Lebesgue measurable? If so, is it a Borel set?