Definition:Additive Function (Measure Theory)
This page is about additive functions in measure theory. For other uses, see Definition:Additive Function.
sections
It may also need to be brought up to our standard house style.
If you have done this or know it has been done, please remove {{Tidy}} from the code.
Definition
Let $\mathcal S$ be an algebra of sets.
Let $f: \mathcal S \to \overline{\R}$ be a function, where $\overline{\R}$ denotes the set of extended real numbers.
Then $f$ is defined to be additive iff:
- $\forall S, T \in \mathcal S: S \cap T = \varnothing \implies f \left({S \cup T}\right) = f \left({S}\right) + f \left({T}\right)$
That is, for any two disjoint elements of $\mathcal S$, $f$ of their union equals the sum of $f$ of the individual elements.
Note from Finite Union of Sets in Additive Function that:
- $\displaystyle f \left({\bigcup_{i \mathop = 1}^n S_i}\right) = \sum_{i \mathop = 1}^n f \left({S_i}\right)$
where $S_1, S_2, \ldots, S_n$ is any finite collection of pairwise disjoint elements of $\mathcal S$.
Such a function is also referred to as a finitely additive function to distinguish it, when necessary, from a countably additive function.
Context
This definition is usually made in the context of measure theory, but the concept reaches a wider field than that.
