Content

A content $μ$ is a is a real-valued function defined on a collection $A$ such that:

$μ (A) \in [0, \infty]$
$μ (\emptyset) = 0$
Let $A_{1}, \dots A_{n} \in A$ , $⋃_{i = 1}^{n} A_{i} \in A$ , and $A_{i} \cap A_{j} = \emptyset$ when $i \neq j$ $$\mu\left(\bigcup_{i = 1}^n A_i\right) = \sum_{i = 1}^n A_i$$
We see that contents are finitely additive. Sometimes it is called a finitely additive measure, in contrast that measures are $σ$ -additive.

We choose $A$ we want it to be a ring of sets, or maybe a semiring of sets

Properties

If $A$ is a semiring of sets then the following statements can be deduced:

Every content $μ$ is monotone that is; $A \subseteq B$ , then $μ (A) \leq μ (B)$ for $A, B \in A$
Every content $μ$ is subadditive that is; $μ (A \cup B) \leq μ (A) + μ (B)$ for $A, B \in A$

If $A$ is a ring of sets one gets additionally:

Substractivity: for $B \subseteq A$ satisfying that $μ (B) < \infty$ it follows that $μ (A ∖ B) = μ (A) - μ (B)$
$A, B \in A$ , then $μ (A \cup B) + μ (A \cap B) = μ (A) + μ (B)$
Subadditivity: $A_{i} \in A$ , with $i \in {1, \dots, n}$ then $$\mu\left(\bigcup_{i = 1}^n A_i\right) \le \sum_{i = 1}^n A_i$$
$σ$ -Superadditivity: For $A_{i} \in A$ , with $i \in N$ , pairwise disjoint satisfying that $⋃_{i \in N} A_{i} \in A$ , then $$\mu\left(\bigcup_{i = 1}^\infty A_i\right) \ge \sum_{i = 1}^\infty A_i $$
If $μ$ is a finite content that is, $A \in A$ , then $μ (A) < \infty$ , then the inclusion-exclusion principle applies: $$ \mu\left(\bigcup_{i=1}^nA_i\right) = \sum_{k=1}^n(-1)^{k+1}!!\sum_{I\subseteq{1,\dotsc,n},\atop |I|=k}!!!!\mu\left(\bigcap_{i\in I}A_i\right)$$where $A_{i} \in A$ for $i \in {1, \dots, n}$ .