 Solution

The products are nonnegative and measurable. independence and the unit means give $\mathbb{E}{Y}_n={\prod }_{k\le n}\mathbb{E}{\xi }_k=1$, so they are integrable.

Let ${\mathcal{H}}_n=\sigma (Y_0,\dots ,Y_n)$ be the natural filtration. It is contained in $\sigma ({\xi }_1,\dots ,{\xi }_n)$, so ${\xi }_{n+1}$ is independent of ${\mathcal{H}}_n$ even if some products have vanished. Using the bounded or integrable nonnegative conditional-product identity,Thus $Y$ is a martingale in its natural filtration.

Doob's convergence theorem in the form needed here states that a nonnegative supermartingale converges almost surely to a finite integrable limit; more generally a submartingale with uniformly bounded first absolute moments has such an almost sure convergence. Applying it to $Y$ givesThe expected value bound follows from Fatou's lemma. The theorem does not assert $L^1$ convergence or equality of expected values without a uniform-integrability hypothesis.