 Product martingale from independent mean-one factors

If $Y_1,Y_2,\dots$ are independent integrable random variables with $\mathbb{E}[Y_i]=1$, thenis a martingale for ${\mathcal{F}}_n=\sigma (Y_1,\dots ,Y_n)$, because independence gives $\mathbb{E}[M_{n+1}\mid {\mathcal{F}}_n]=M_n\mathbb{E}[Y_{n+1}]=M_n$.