Content (measure theory)

In mathematics, in particular in measure theory, a content is a real-valued function defined on a collection of subsets such that That is, a content is a generalization of a measure: while the latter must be countably additive, the former must only be finitely additive. In many important applications the is chosen to be a ring of sets or to be at least a semiring of sets in which case some additional properties can be deduced which are described below. For this reason some authors prefer to define contents only for the case of semirings or even rings. If a content is additionally σ-additive it is called a pre-measure and if furthermore is a σ-algebra, the content is called a measure. Therefore every (real-valued) measure is a content, but not vice versa. Contents give a good notion of integrating bounded functions on a space but can behave badly when integrating unbounded functions, while measures give a good notion of integrating unbounded functions. A classical example is to define a content on all half open intervals by setting their content to the length of the intervals, that is, One can further show that this content is actually σ-additive and thus defines a pre-measure on the semiring of all half-open intervals. This can be used to construct the Lebesgue measure for the real number line using Carathéodory's extension theorem. For further details on the general construction see article on Lebesgue measure. An example of a content that is not a measure on a σ-algebra is the content on all subsets of the positive integers that has value on any integer and is infinite on any infinite subset. An example of a content on the positive integers that is always finite but is not a measure can be given as follows. Take a positive linear functional on the bounded sequences that is 0 if the sequence has only a finite number of nonzero elements and takes value 1 on the sequence so the functional in some sense gives an "average value" of any bounded sequence. (Such a functional cannot be constructed explicitly, but exists by the Hahn–Banach theorem.