a) Since F is continuous, it takes compact sets to compact sets, hence takes $ F_\sigma $ sets to $ F_\sigma $ sets. By absolute continuity, F takes sets of measure zero to sets of measure zero. Done.
b) Let $ [a,b] \subset [0,1] $. F is continuous, hence $ F[a,b] = [c,d] $, and $ \exists \ \alpha, \beta, F(\alpha) = c, F(\beta) = d $
$ \Rightarrow |F[a,b]| = |\int_\alpha^\beta f| \leq \int_a^b |f| $.
Hence it's true for compact sets, and by extension, for $ F_\sigma $ sets, and since F takes sets of measure zero to sets of measure zero, it's true for all measurable sets.
