Axiom:Content Axioms

Definition
Let $X$ be a set.

Let $C \subset \powerset X$ be a set of subsets of $X$.

Let $\lambda : C \to \R_{\mathop \ge 0}$ be a function from $C$ to the non-negative real numbers.

Then $\lambda$ is a content on $C$, for any $A,B \in C$:
 * $(1): \quad A \subset B \implies \map \lambda A \le \map \lambda B$
 * $(2): \quad A \cap B = \empty \implies \map \lambda {A \cup B} = \map \lambda A + \map \lambda B$
 * $(3): \quad \map \lambda {A \cup B} \le \map \lambda A + \map \lambda B$