Product martingale from independent mean-one factors
= Product martingale from independent mean-one factors
If $Y_1,Y_2,\ldots$ are independent integrable random variables with $\mathbb E[Y_i]=1$, then
$$
M_n=\prod_{i=1}^nY_i
$$
is a martingale for $\mathcal F_n=\sigma(Y_1,\ldots,Y_n)$, because independence gives $\mathbb E[M_{n+1}\mid\mathcal F_n]=M_n\mathbb E[Y_{n+1}]=M_n$.