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{{MSC|28A}}

[[Category:Classical measure theory]]

{{TEX|done}}

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 of sets|ring]] (more often an [[Algebra of sets|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 [[Ring of sets|$\sigma$-ring]] (more often a [[Algebra of sets|$\sigma$-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|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|probability measure]]. $\sigma$-additive set functions taking values in the extended real line $[-\infty, \infty]$ are commonly called [[Signed measure|signed measures]] (some authors use also the name ''charge''), whereas $\sigma$-additive set functions taking values in vector spaces are commonly called [[Vector measure|vector measures]].

*Outer measures. The domain of definition $\mathcal{S}$ of an [[Outer measure|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====
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|valign="top"|{{Ref|Ha}}|| P.R. Halmos, "Measure theory" , v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}}
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|valign="top"|{{Ref|DS}}||  N. Dunford, J.T. Schwartz, "Linear  operators. General theory" , '''1'''  , Interscience (1958)  {{MR|0117523}}
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|}