**Exercise.*** Let be a filtered probability space that satisfies the usual conditions. We denote*

and for , is the restriction of to . Let be a probability measure on such that for every ,

- Show that there exists a right continuous and left limited martingale such that for every ,
- Show that the following properties are equivalent:

1) ;

2) The martingale is uniformly integrable;

3) converges in ;

4) almost surely converges to an integrable and measurable random variable such that