TITLE: Does "All additive functions are linear" imply every set of reals is measurable?
QUESTION [22 upvotes]: Suppose we are working in ZF set theory without choice. An additive function is a function defined over the real line such that $f(x+y)=f(x)+f(y)$. It is known that if every set of reals is measurable, then every additive function is linear. Is the converse true in ZF?

REPLY [16 votes]: No.
We know that "Every set of reals has the Baire property" will also have the same consequence, and that ZF+"Every set of reals has the Baire property" is actually a weaker(!) theory than ZF+"Every set of reals is Lebesgue measurable", as shown by Shelah.