Axiom:Content Axioms

Definition

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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 $C$ if and only if, for any $A, B \in C$:

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\((1)\)	$:$		Monotone	\(\ds \forall A, B \in C:\)	\(\ds A \subseteq B \implies \map \lambda A \le \map \lambda B \)
\((2)\)	$:$		Additive	\(\ds \forall A, B \in C:\)	\(\ds A \cap B = \O \implies \map \lambda {A \cup B} = \map \lambda A + \map \lambda B \)
\((3)\)	$:$		Subadditive	\(\ds \forall A, B \in C:\)	\(\ds \map \lambda {A \cup B} \le \map \lambda A + \map \lambda B \)

Sources

• 1950: Paul R. Halmos: Measure Theory: $\S 53$: Generation of Borel Measures