Greedy Algorithm guarantees Maximum Weight iff Matroid

Theorem
Let $S$ be a finite set.

Let $\mathscr I$ be a non-empty set of subsets of $S$.

Then $\mathscr I$ is the set of independent subsets of a matroid on $S$ :
 * $(1) \quad \struct{S, \mathscr I}$ is an independence system
 * $(2) \quad$ for all non-negative weight functions $w : S \to \R_{\ge 0}$, the Greedy Algorithm selects $A_w \in \mathscr I$:
 * $\forall B \in \mathscr I: \map {w^+} {A_w} \ge \map {w^+} B$
 * where $w^+$ denotes the extended weight function of $w$.