Let us first remind some basic facts about the notion of uniform integrability which is a crucial tool in the study of continuous time martingales.

**Definition.** * Let be a family of random variables. We say that the family is uniformly integrable if for every , there exists such that*

We have the following properties:

- A finite family of integrable random variables is uniformly integrable ;
- If the family is uniformly integrable then it is bounded in , that is ;
- If the family is bounded in with , that is , then it is uniformly integrable.

The notion of uniform integrability is often used to prove a convergence in thanks to the following result:

**Proposition.** * Let be a sequence of integrable random variables. Let be an integrable random variable. The sequence converges toward in , that is , if and only if:*

- In probability, , that is for every , ;
- The family is uniformly integrable.

It is clear that if is an integrable random variable defined on a filtered probability space then the process is a martingale with respect to the filtration . The following theorem characterizes the martingales that are of this form.

**Theorem (Doob’s convergence theorem):**

*Let be a filtration defined on a probability space and let be a martingale with respect to the filtration whose paths are left limited and right continuous. The following properties are equivalent:*

- When , converges in ;
- When , converges almost surely toward an integrable and -measurable random variable that satisfies
- The family is uniformly integrable.

**Proof:**

As a first step, we show that if the martingale is bounded in , that is

then almost surely converges toward an integrable and -measurable random variable .

Let us first observe that

Therefore, in order to show that almost surely converges when , we may prove that

Let us assume that

In that case we may find such that:

The idea now is to study the oscillations of between and . For , and , we denote

and

Let be the greatest integer for which we may find elements of ,

that satisfy

Let now

where is recursively defined by:

Since is martingale, it is easily checked that

Furthermore, thanks to the very definition of , we have

Therefore,

and thus

This implies that almost surely , from which we deduce

Since the paths of are right continuous, we have

This is absurd.

Thus, if is bounded in , it almost surely converges toward an -measurable random variable . Fatou’s lemma provides the integrability of .

With this preliminary result in hands, we can now turn to the proof of the theorem.

Let us assume that converges in . In that case, it is of course bounded in , and thus almost surely converges toward an -measurable and integrable random variable . Let and , we have for ,

By letting , the dominated convergence theorem yields

Therefore, as expected, we obtain

Let us now assume that almost surely converges toward an -measurable and integrable random variable that satisfies

We almost surely have and thus for ,

This implies the uniform integrability for the family .

Finally, if the family is uniformly integrable, then it is bounded in and therefore almost surely converges. The almost sure convergence, together with the uniform integrability, provides the convergence in .