Suppose that $U$ and $V$ are random variables. Then I want to show that if $ E[g(U)\mid V]=g(V), $ for all bounded non-negative Borel-measureable functions $g:\mathbb{R}\to\mathbb{R}_+$, then $U=V$ a.s.
I'm given a hint, which is to use the following result:
If $X\in \mathcal{L}^2(P)$, $\mathcal{B}$ is a sub-$\sigma$-field and we put $Y=E[X\mid \mathcal{B}]$ then $ X\sim Y \Rightarrow X=Y \,\text{ a.s.} $
My thoughts so far: If I can show that $g(U)\sim g(V)$ for all such $g$, then $g(U)=g(V)$ a.s., and I think this is enough to show that $U=V$ a.s. (i.e. I'm thinking of using $g_n(x)=-n\vee x\wedge n$ and let $n\uparrow \infty$). Please correct me if I'm wrong here. If this the way to proceed, I just need a hint or a tip on how to show $g(U)\sim g(V)$.
Thanks in advance.