Let $S$ be a bounded set in $\mathbb{R}^n$ that is the union of the countable collection of rectifiable sets $S_1, S_2, \dots$.
How can we show that $S_1 \bigcup \dots \bigcup S_n$ is rectifiable? Also, is there an example showing that S need not be rectifiable?
My proof:
From the definition of rectifiable, if we look at a bounded set S in R^n, then whose volume over S of the constant function 1 is integrable.
Also, A subset S of R^n is rectifiable iff S is bounded and Bd S has measure Zero.
So, we are told that S is bounded and we need to show that Bd(S) is zero. For S is bounded, and Bd(S) = Reals, which has measure zero.