Let the function $f:[a,b] \to \mathbb R$ be Lipschitz, that is, there is a constant $c \geq 0$ such that for all $u,v \in [a,b]$, $|f(u)-f(v)| \leq c|u-v|$. Show that $f$ maps a set of measure zero onto a set of measure zero. Show that $f$ maps an $F_\sigma$ set onto an $F_\sigma$ set. Conclude that $f$ maps a measurable set to a measurable set.
I have a question about this. We know that a set of measure zero must either be of the form , or it must be a set of countable elements. But for this function, we know that the domain is an interval...and we cannot have an interval with only countable elements, because an interval contains all reals, right? So for the first part of this question, we need to prove that  is mapped to , right?
Thanks in advance