Content (measure theory)
In mathematics, a content is a set function like a measure but a content need not be countably additive, but must only be finitely additive. A content is a real function defined on a field of sets such that
An example of a content is a measure, which is a σ-additive content defined on a σ-field. 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.
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 1/2n on the integer n 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 1, 1, 1, ...., 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.) Then the content of a set of positive integers is the average value of the sequence that is 1 on this set and 0 elsewhere. Informally, one can think of the content of a subset of integers as the "chance" that a randomly chosen integer lies in this subset (though this is not compatible with the usual definitions of chance in probability theory, which assume countable additivity).
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