Solution
= Solution
The variable $M_n$ is $\mathcal F_n$-measurable. Positivity, independence, and the mean-one assumption give
$$
\mathbb E[M_n]=\prod_{i=1}^n\mathbb E[Y_i]=1,
$$
so it is integrable. Since $Y_{n+1}$ is independent of $\mathcal F_n$,
$$
\mathbb E[M_{n+1}\mid\mathcal F_n]
=\mathbb E[M_nY_{n+1}\mid\mathcal F_n]
=M_n\mathbb E[Y_{n+1}]
=M_n.
$$
Thus \b[$(M_n)$ is the <product martingale from independent mean-one factors>.]