Complete Family of pdf/pmf

Definition

Let:

• $Z$ : Random variable
• pdf/pmf of $Z$ is one member of the family $\{h(z;\theta ):\theta \in \Omega \}$

If $\forall \theta \in \Omega :E[u(Z)]=0⟹u(z)=0$ almost surely with respect to $h(z;\theta ),\forall \theta \in \Omega$

Then the family $h(z;\theta ):\theta \in \Omega$ is called a complete family of pdf/pmfs.

Remark

In short, If $E[u(Z)]=0$ for all $\theta$ implies $u(Z)=0$ (almost surely), then the family is complete.

“Almost surely” means “with probability 1” or equivalently “except possibly on a set of probability zero.”

The if statement can be reworded as “If the condition $E[u(Z)]=0$ for every $\theta \in \Omega$ requries $u(z)=0$, except on a set of points that has probability zero for each $h(z;\theta )$, $\theta \in \Omega$”

In practical terms, if something happens almost surely, it might technically fail on some points, but those points have zero probability of occurring.