# Axiom:Content Axioms

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## 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