2a.
$ ( \Rightarrow ) $ Say $ f $ is A.C. Then $ f $ is of bounded variation, and since $ f $ is clearly nondecreasing, $ f $ must be bounded.
In particular, $ \infty > f(1)=\sum_{n=1}^\infty m([0,1]\cap G_n) \Rightarrow \sum_{n=1}^\infty m(G_n)<\infty $.
$ ( \Leftarrow ) $
