Editing Measure (mathematics) (section)

==Generalizations==
For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity. For instance, a countably additive [[set function]] with values in the (signed) real numbers is called a ''[[signed measure]]'', while such a function with values in the [[complex numbers]] is called a ''[[complex measure]]''. Observe, however, that complex measure is necessarily of finite [[total variation|variation]], hence complex measures include [[Finite measure|finite signed measures]] but not, for example, the [[Lebesgue measure]].

Measures that take values in [[Banach spaces]] have been studied extensively.<ref>{{citation
 | last = Rao | first = M. M.
 | isbn = 978-981-4350-81-5
 | mr = 2840012
 | publisher = [[World Scientific]]
 | series = Series on Multivariate Analysis
 | title = Random and Vector Measures
 | volume = 9
 | year = 2012}}.</ref> A measure that takes values in the set of self-adjoint projections on a [[Hilbert space]] is called a ''[[projection-valued measure]]''; these are used in [[functional analysis]] for the [[spectral theorem]]. When it is necessary to distinguish the usual measures which take non-negative values from generalizations, the term '''positive measure''' is used. Positive measures are closed under [[conical combination]] but not general [[linear combination]], while signed measures are the linear closure of positive measures. More generally see [[measure theory in topological vector spaces]].

Another generalization is the ''finitely additive measure'', also known as a [[Content (measure theory)|content]]. This is the same as a measure except that instead of requiring ''countable'' additivity we require only ''finite'' additivity. Historically, this definition was used first. It turns out that in general, finitely additive measures are connected with notions such as [[Banach limit]]s, the dual of [[lp space|<math>L^\infty</math>]] and the [[Stone–Čech compactification]]. All these are linked in one way or another to the [[axiom of choice]].  Contents remain useful in certain technical problems in [[geometric measure theory]]; this is the theory of [[Banach measure]]s.

A ''charge'' is a generalization in both directions: it is a finitely additive, signed measure.<ref>{{Cite book|last=Bhaskara Rao|first=K. P. S.|url=https://www.worldcat.org/oclc/21196971|title=Theory of charges: a study of finitely additive measures|date=1983|publisher=Academic Press|others=M. Bhaskara Rao|isbn=0-12-095780-9|location=London|pages=35|oclc=21196971}}</ref> (Cf. [[ba space]] for information about ''bounded'' charges, where we say a charge is ''bounded'' to mean its range its a bounded subset of ''R''.)