is a real-valued function defined on a collection of subsets
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
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
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 on the positive integers that is always finite but is not a measure can be given as follows.
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.)
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).
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.
forms a Semi ring of sets then the following statements can be deduced: If furthermore
is a Ring of sets one gets additionally: 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.
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
form a finite collections of disjoint half-open sets whose union covers the range of
form a Banach space with respect to the supremum norm.
The positive elements of the dual of this space correspond to bounded contents
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.
There are several ways to construct a measure μ from a content
This section gives one such method for locally compact Hausdorff spaces such that the content is defined on all compact subsets.
An example is given by the construction of Haer measure on a locally compact group.
One method of constructing such a Hear measure is to produce a left-invariant function
as above on the compact subsets of the group, which can then be extended to a left-invariant measure.
Given λ as above, we define a function μ on all open sets by