Set function

2020 Mathematics Subject Classification: Primary: 28A [MSN][ZBL]

A mapping $\mu$ defined on a family $\mathcal{S}$ of subsets of a set $X$. Commonly the target of $\mu$ is a topological vector space $V$ (more generally a commutative topological group) or the extended real line $[-\infty, \infty]$ (in the latter case, to avoid operations of type $\infty + (-\infty)$ it is assumed that the range is either contained in $[-\infty, \infty[$ or in $]-\infty, \infty]$). It is usually assumed that the empty set is an element of $\mathcal{S}$ and that $\mu (\emptyset) =0$.

Notable examples are

• Finitely additive set functions. In this case the domain of definition is a ring (more often an algebra) and $\mu$ has the property that

\[ \mu \left(\bigcup_{i=1}^N E_i\right) = \sum_{i=1}^N \mu (E_i) \] for every finite collection $\{E_i\}$ of disjoint elements of $\mathcal{S}$.

• Measures. In this case the domain of definition $\mathcal{S}$ is a $\sigma$-ring (more often a $\sigma$-algebra) and the set function is assumed to be $\sigma$-additive (or, equivalently countably additive), that is

\[ \mu \left(\bigcup_{i=1}^\infty E_i\right) = \sum_{i=1}^\infty \mu (E_i) \] for every countable collection $\{E_i\}$ of disjoint elements of $\mathcal{S}$. Note that, since we assume $\mu (\emptyset) = 0$, a measure is always finitely additive.

The word measure is indeed commonly used for such set functions which are taking values in $[0, \infty]$ and if in addition $\mu (X)=1$, then $\mu$ is a probability measure. $\sigma$-additive set functions taking values in the extended real line $[-\infty, \infty]$ are commonly called signed measures (some authors use also the name charge), whereas $\sigma$-additive set functions taking values in vector spaces are commonly called vector measures.

• Outer measures. The domain of definition $\mathcal{S}$ of an outer measure $\mu$ is an hereditary $\sigma$-ring (also called $\sigma$-ideal), i.e. a $\sigma$-ring $\mathcal{S}$ with the additional property that it contains any subset of any of its elements (however, the most commonly used outer measures are defined on the whole space $\mathcal{P} (X)$ of all subsets of $X$). An outer measure takes values in $[0, \infty]$ and it is required to be $\sigma$-subadditive (or countably subadditive), i.e.

\[ \mu \left(\bigcup_{i=1}^\infty E_i\right) \leq \sum_{i=1}^\infty \mu (E_i) \] for every countable collection $\{E_i\}$ of subsets of $X$.

References

[Ha]	P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802
[DS]	N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523