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 $[-\mathrm{\infty },\mathrm{\infty }]$ (in the latter case, to avoid operations of type $\mathrm{\infty }+(-\mathrm{\infty })$ it is assumed that the range is either contained in $[-\mathrm{\infty },\mathrm{\infty }[$ or in $]-\mathrm{\infty },\mathrm{\infty }]$). It is usually assumed that the empty set is an element of $\mathcal{S}$ and that $\mu (\mathrm{\varnothing })=0$.

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

$$\mu \left(\bigcup _{i=1}^{\mathrm{\infty }}{E}_i\right)=\sum _{i=1}^{\mathrm{\infty }}\mu (E_i)$$ for every countable collection $\{E_i\}$ of disjoint elements of $\mathcal{S}$. Note that, since we assume $\mu (\mathrm{\varnothing })=0$, a measure is always finitely additive.

The word measure is indeed commonly used for such set functions which are taking values in $[0,\mathrm{\infty }]$ and if in addition $\mu (X)=1$, then $\mu$ is a probability measure. $\sigma$-additive set functions taking values in the extended real line $[-\mathrm{\infty },\mathrm{\infty }]$ 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 $[0,\mathrm{\infty }]$ and it is required to be $\sigma$-subadditive (or countably subadditive), i.e.

$$\mu \left(\bigcup _{i=1}^{\mathrm{\infty }}{E}_i\right)\le \sum _{i=1}^{\mathrm{\infty }}\mu (E_i)$$ for every countable collection $\{E_i\}$ of subsets of $X$.

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