**Definitions:**

Two real valued random variables $X_0$ and $X_1$ are called a *one step martingale* if $E[X_1| X_0] = X_0$. 

We say the one step martingale is in $L^2$ if both $X_0$ and $X_1$ are in $L^2(P)$.

**Question:**

Given an $L^2$ one step martingale $(X_0, X_1)$ does there exist a sequence $(Y_0^n, Y_1^n)$ of $L^2$ one step martingales satisfying the following two conditions?

- $Y_0, Y_1$ are simple random variables, i.e. variables taking only a finite number of values.
 - $Y_0^n, Y_1^n$ converge to $X_0, X_1$ respectively in $L^2$.