Example of a martingale with conditions

Question

I'm going through Klenke's textbook (Ex 11.2.1) and there's an exercise asking to think of a martingale (which is nonnegative) $M$ where $\mathbb{E}(M_n) = 1, \forall n \in \mathbb N$ where $\lim_{n\to \infty} M_n =0$ almost surely.

How would i approach this type of question and think of an example? I can't seem to think of one with these conditions.

Answer 1

There is a well known example. Consider $(0,1)$ with Lebesgue measure and define $M_n=nI_{(0,\frac 1 n)}$. You can show that this is martingale with the desired properties.