Conditional Expectations of Integrable Random Variable with respect to Filtration forms Martingale/Continuous Time

Theorem
Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a continuous-time filtered probability space.

Let $Z$ be an integrable random variable.

For each $t \in \hointr 0 \infty$, let $\expect {Z \mid \FF_t}$ be a version of the conditional expectation of $Z$ given $\FF_t$.

For each $t \in \hointr 0 \infty$ set:


 * $X_t = \expect {Z \mid \FF_t}$

Then $\sequence {X_t}_{t \ge 0}$ is a $\sequence {\FF_t}_{t \ge 0}$-martingale.

Proof
From the definition of the conditional expectation of $Z$ given $\FF_t$, we have that:


 * $X_t$ is $\FF_t$-measurable for each $t \in \hointr 0 \infty$.

So $\sequence {X_t}_{t \ge 0}$ is $\sequence {\FF_t}_{t \ge 0}$-adapted.

Now let $s, t \in \hointr 0 \infty$ have $s \le t$.

Then we have:

So $\sequence {X_t}_{t \ge 0}$ is a $\sequence {\FF_t}_{t \ge 0}$-martingale.