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 forms a Semiring of sets then the following statements can be deduced: Every content is monotone that is, Every content is subadditive that is, for such that On rings If furthermore is a Ring of sets one gets additionally: Subtractivity: for satisfying it follows Subadditivity: -Superadditivity: For any pairwise disjoint satisfying we have If is a finite content, that is, then the inclusion–exclusion principle applies: where for all 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 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 with respect to the content as where the form a finite collections of disjoint half-open sets whose union covers the range of and is any element of and where the limit is taken as the diameters of the sets tend to 0. Duals of spaces of bounded functions Suppose that is a measure on some space The bounded measurable functions on form a Banach space with respect to the supremum norm. The positive elements of the dual of this space correspond to bounded contents with the value of on given by the integral 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 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