# Axiom:Content Axioms

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

- $(1): \quad A \subset B \implies \map \lambda A \le \map \lambda B$
- $(2): \quad A \cap B = \empty \implies \map \lambda {A \cup B} = \map \lambda A + \map \lambda B$
- $(3): \quad \map \lambda {A \cup B} \le \map \lambda A + \map \lambda B$

## Sources

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- 1974: Paul R. Halmos:
*Measure Theory*