almost everywhere as defined by an ultrafilter. The definition of almost everywhere in terms of ultrafilters is closely related to the definition in terms of measures, because each ultrafilter defines a finitely-additive measure taking only the values 0 and 1, where a set has measure 1 if and only if it is included in the ultrafilter. See also Dirichlet's function, a function that is equal to 0 almost everywhere. References Bibliography Measure theory Mathematical terminology ja:ほとんど (数学)#ほとんど至るところで