TITLE: What are some reasonable-sounding statements that are independent of ZFC?
QUESTION [272 upvotes]: Every now and then, somebody will tell me about a question. When I start thinking about it, they say, "actually, it's undecidable in ZFC."
For example, suppose $A$ is an abelian group such that every short exact sequence of abelian groups $0\to\mathbb Z\to B\to A\to0$ splits. Does it follow that $A$ is free? This is known as Whitehead's Problem, and it's undecidable in ZFC.
What are some other statements that aren't directly set-theoretic, and you'd think that playing with them for a week would produce a proof or counterexample, but they turn out to be undecidable? One answer per post, please, and include a reference if possible.

REPLY [4 votes]: "There exists a complete metric space $(X,d)$ and a Borel probability measure $\mu$ on $X$ with non-separable support."
This has been discussed elsewhere on MO (I forget where, sorry), but is shown to require a large cardinal axiom in Fremlin's Measure Theory, section 438.