I am stuck on the following problem: Let $X_t$ be a stochastic process. We have that for all $\varphi$ that are continuous and bounded it holds:
$$E[\varphi(X_t)|\mathcal{F}_t] = \varphi(E[X_t| \mathcal{F}_t]) $$
where $\mathcal{F}_t$ it is some filtration. Then it follows that $X_t$ is $\mathcal{F}_t$ adapted. I am pretty sure that I've already seen this result somewhere but at the moment I cannot figure out where. May be someone here can give me a hint or a reference.
Because $t$ is fixed and will play no role, I'll drop it from the story. Thus we have an integrable random variable $X$ such that $E[\varphi(X)|\mathcal F]=\varphi(E[X|\mathcal F])$ for all bounded and continuous $\varphi$. I will allow $\varphi$ to be complex-valued. (Look at the real and imaginary parts of $\varphi$ separately and then add.) Let's define $Y:=E[X|\mathcal F]$. Now take $\varphi(x) = \exp(i\lambda x)$ in the basic identity, multiply both sides by $\exp(-i\lambda Y)$, and then take expectations. There results $$ E[\exp(i\lambda(X-Y))]=E[\exp(i\lambda Y)\exp(-i\lambda Y)]=1, $$ which is true for all real $\lambda$. What does this tell you about the random variable $X-Y$?