Absolute continuity preserves measurability In studying absolutely continuous function, I knew that if $f:[a,b]\to\mathbb{R}$ is absolutely continuous, then $f(N)$ has measure zero if $N$ is, and $f(E)$ is measurable if $E$ is.
Suppose continuous function $f:[a,b]\to\mathbb{R}$ is such that if $E$ is measurable, then $f(E)$ is measurable. Then does it follow that if $N$ has measure zero, then $f(N)$ has measure zero?

Suppose $N\subset[a,b]$ is of measure $0$, but $f(N)$ is not. Then, as $f(N)$ is measurable, it contains a non-measurable set $B$. One can then find a subset $A$ of $N$ with $f(A)=B$. As $A$ is measurable with measure $0$, we have a contradiction.
In fact, a continuous function on $[a,b]$ maps every measurable set onto a measurable set if and only if it maps measure zero sets to measure zero sets. This is Exercise 18.39 b) in Hewitt and Stromberg's Real and Abstract Analysis.
