Definitions:

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

Question:

Given a one step martingale $(X_0, X_1)$ does there exist a sequence $(Y_0^n, Y_1^n)$ of 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 $L^2$.

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 $L^2$.