Content (measure theory)
In mathematics, a content is a set function that is like a measure, but a content must only be finitely additive, whereas a measure must be countably additive. A content is a real function formula_1 defined on a collection of subsets formula_2 such that In many important applications the formula_2 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 formula_2 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. Examples. A classical example is to define a content on all half open intervals formula_8 by setting their content to the length of the intervals, that is, formula_9 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 formula_10 on any integer formula_11 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 formula_12 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). Properties. Frequently contents are defined on collections of sets that satisfy further constraints. In this case additional properties can be deduced that fail to hold in general for contents defined on any collections of sets. On semirings. If formula_2 forms a Semiring of sets then the following statements can be deduced: On rings. If furthermore formula_2 is a Ring of sets  one gets additionally: Integration of bounded functions. In general integration of functions with respect to a content does not behave well. However there is a well-behaved notion of integration provided that the function is bounded and the total content of the space is finite, given as follows. Suppose that the total content of a space is finite. If formula_35 is a bounded function on the space such that the inverse image of any open subset of the reals has a content, then we can define the integral of formula_35 with respect to the content as formula_37 where the formula_38  form a finite collections of disjoint half-open sets whose union covers the range of formula_39 and formula_40 is any element of formula_41 and where the limit is taken as the diameters of the sets formula_38 tend to 0. Duals of spaces of bounded functions. Suppose that formula_1 is a measure on some space formula_44 The bounded measurable functions on formula_45 form a Banach space with respect to the supremum norm. The positive elements of the dual of this space correspond to bounded contents formula_46 formula_47 with the value of formula_46 on formula_35 given by the integral formula_50 Similarly one  can form the space of essentially bounded functions, with the norm given by the essential supremum, and the positive elements of the dual of this space are given by bounded contents that vanish on sets of measure 0. Construction of a measure from a content. There are several ways to construct a measure μ from a content formula_46 on a topological space. This section gives one such method for locally compact Hausdorff spaces such that the content is defined on all compact subsets. In general the measure is not an extension of the content, as the content may fail to  be countably additive, and the measure may even be identically zero even if the content is not. First restrict the content to compact sets. This gives a function formula_46 of compact sets formula_53 with the following properties: There are also examples of functions formula_46 as above not constructed from contents. An example is given by the construction of Haar measure on a locally compact group. One method of constructing such a Haar measure is to produce a left-invariant function formula_46 as above on the compact subsets of the group, which can then be extended to a left-invariant measure . Definition on open sets. Given λ as above, we define a function μ on all open sets by formula_62 This has the following properties: Definition on all sets. Given μ as above, we extend the function μ to all subsets of the topological space by formula_68 This is an outer measure, in other words it has the following properties: Construction of a measure. The function μ above is an outer measure on the family of all subsets. Therefore it becomes a measure when restricted to the measurable subsets for the outer measure, which are the subsets formula_73 such that formula_74  for all subsets formula_44 If the space is locally compact then every open set is measurable for this measure. The measure formula_1 does not necessarily coincide with the content formula_46 on compact sets, However it does if formula_46 is regular in the sense that for any compact formula_79 formula_80 is the inf of formula_81 for compact sets formula_82 containing formula_53 in their interiors.