Possibly related: Show rigorously that Pólya urn describes a martingale
Suppose we had $b$ and $r$ blue/red balls in urn at time $0$, and at each $n\ge 1$ we draw a ball randomly and then put it back with another ball of the same color. Let $B_n$ and $R_n$ denote blue/red balls in urn at time $n$. Let
$$ X_n=\frac{B_n}{B_n+R_n}=\frac{B_n}{b+r+n} $$
i.e. fraction of blue balls in the urn at time $n$. Given i.i.d uniform random variables $U_n$ in $[0,1]$, then we define for all $n\ge 0$
$$ B_{n+1}=(B_n+1)1_{U_n\le X_n}+B_n1_{U_n>X_n} $$
and similarly for $R_n$.
Suppose we want to show that $X_n$ is a martingale with respect to $F_n=\sigma(U_1,\cdots,U_n)$ and $F_0=\{\emptyset,\Omega\}$. It's trivial that $X_n$ are integrable and measurable w.r.t $F$.
However, I'm having hard time showing that $E[X_{n+1}| F_n]=X_n$. The problem boils down to the fact that with given definition of $B_{n+1}$ and $R_{n+1}$, both of them are measurable with respect to $F_n$, and in fact this would imply $X_n$ is martingale iff $X_{n+1}=X_n$ for all $n$, which is a nonsense.
Is it just this exercise being wrongly stated (indeed if $R_n$ and $B_n$ are determined by $U_{n+1}$ then we would have $X_n$ martingale) or am I missing something?
