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 of compact sets with the following properties: for all compact sets for all pairs of compact sets for all pairs of disjoint compact sets. There are also examples of functions 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 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 This has the following properties: for any collection of open sets for any collection of disjoint open sets. Definition on all sets Given μ as above, we extend the function μ to all subsets of the topological space by This is an outer measure, in other words it has the following properties: for any countable collection of sets. 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 such that for all subsets If the space is locally compact then every open set is measurable for this measure. The measure does not necessarily coincide with the content on compact sets, However it does if is regular in the sense that for any compact is the inf of for compact sets containing in their interiors. See also References Measure theory Families of sets