The general rectangle criterion agrees with the published finite random-variable definition

By [L2] and [L1], the variables $X_i$ are genuine random elements on the full-power-set probability space, so [L3] applies to them.

Conversely, assume [L4]. For a finite subfamily $(X_i{)}_{i\in J}$ and measurable sets $B_i\subseteq \mathbb{R}$, only finitely many values in $B_i\cap X_i(\mathrm{\Omega })$ can occur. The event $\{X_i\in B_i\text{ for all }i\in J\}$ is the disjoint union of the attained-value events $\{X_i=x_i\text{ for all }i\in J\}$ over those finitely many tuples. Summing the factorized singleton probabilities from [L4] gives ${\mathbb{P}}_w(X_i\in B_i\text{ for all }i\in J)={\prod }_{i\in J}{\mathbb{P}}_w(X_i\in B_i)\mathrm{.}$ So the rectangle criterion holds.

If the rectangle criterion holds, apply it to singleton target sets $B_i=\{x_i\}$. This gives ${\mathbb{P}}_w(X_i=x_i\text{ for all }i\in J)={\prod }_{i\in J}{\mathbb{P}}_w(X_i=x_i)$ for every finite $J\subseteq I$, which is exactly [L4].

Steps 2.1 and 1.2 prove that the finite published definition and the general rectangle criterion agree exactly on finite-valued variables over a finite probability space.