Why is a Martingale defined as a discrete process where the conditional expected value of the next observation is equal to the previous

Question

In my notes, I am seeing that a Martingale is defined as

A discrete time stochastic process where the conditional expected value of the next observation, given all past observations, is equal to the most recent observation.

or mathematically

$$
E[Z_{n+1}|Z_n = z_n, \ldots, Z_1 = z_1] = z_n
$$

where $Z_n$ is a Martingale.

I am wondering why the statement is not more general because we can show that

$$
E[Z_n] = E[Z_{n – 1}] = \ldots = E[Z_1]
$$

Instead of a conditional expectation, we simply have that the unconditional expectation is equal to the previous expectation.

Answer 1

Consider two properties: $$ A:\qquad E[Z_{n+1}|Z_n = z_n, \ldots, Z_1 = z_1] = z_n $$ and $$ B:\qquad E[Z_n] = E[Z_{n - 1}] = \ldots = E[Z_1] $$ Then $A$ implies $B$, but $B$ does not imply $A$. So, $B$ is a property of martingales; but $A$ is much a more powerful condition.