$\mathbb{E}[X_{n}\mid \mathcal{F}_n]$ does not converge in probability to $0$.

Question

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space.

Give an example of a sequence of positive random variables $\left(X_n\right)_{n\ge 1}$ and a sequence $\left(\mathcal{F}_n\right)_{n\ge 1} $ of sub-tribes of $\mathcal{F}$ such that $X_n$ converges in probability to $0$ and that $\mathbb{E}[X_{n}\mid \mathcal{F}_n]$ does not converge in probability to $0$.

Answer 1

If $\mathcal F_n =\{\emptyset, \Omega\}$ for all $n$ then $E(X|\mathcal F_n)=EX_n$. So all you need is a positive sequence $(X_n)$ with finite mean converging to $0$ in probability with $EX_n$ not converging to $0$. Can you find such an example?

Answer 2

Let $(\Omega,\mathcal{F},\mathbb{P})=([0,1],\mathcal{B}, \lambda)$, with the latter denoting the Borel algebra and the measure $\lambda$.

Let $X_n= n 1_{[0,1/n]}.$ Then, $X_n$ clearly converges to $0$ in probability and is positive and bounded (hence, admits a conditional expectation). Let $\mathcal{F}_n=\{\emptyset, [0,1]\}$ for each $n$. Then, $\mathbb{E}(X_n|\mathcal{F}_n)=\mathbb{E}(X_n)=1,$ which clearly doesn't go to $0$.