Let $A\subset[0,1]$ be measurable, and let $g\in L^2(A,dx)$.
Let $C=\{f\in L^2[0,1]:m\{x\in A:f(x) \ne g(x)\}=0 \}$, that is, the set of functions which are equivalent to $g$ on $A$.
Prove that $C$ is closed an convex and find $C^{\perp}$.
Proving convexity is trivial, I've managed to prove closeness, but in quite a tedious way, if anyone sees an elegant way of doing so I'd like to know.
My "real" question, though, is how to calculate $C^{\perp}$.