| The topos Sh( F ) of sheaves on a sigma-algebra F is a natural home for measure theory. The collection of measures is a sheaf, the collection of measurable real valued functions is a sheaf, the operation of integration is a natural transformation, and the concept of almost-everywhere equivalence is a Lawvere-Tierney topology.;The sheaf of measurable real valued functions is the Dedekind real numbers object in Sh( F ) (Scott), and the topology of "almost everywhere equivalence" is the closed topology induced by the sieve of negligible sets (Wendt). The other elements of measure theory have not previously been described using the internal language of Sh( F ). The sheaf of measures, and the natural transformation of integration, are here described using the internal languages of Sh( F ) and F&d14; , the topos of presheaves on F .;These internal constructions describe corresponding components in any topos E with a designated topology j. In the case where E=L&d14; is the topos of presheaves on a locale, and j is the canonical topology, then the presheaf of measures is a sheaf on L . A definition of the measure theory on L is given, and it is shown that when Sh( F ) ≃ Sh( L ), or equivalently, when L is the locale of closed sieves in F this measure theory coincides with the traditional measure theory of a sigma-algebra F . In doing this, the interpretation of the topology of "almost everywhere" equivalence is modified so as to better reflect non-Boolean settings.;Given a measure mu on L , the Lawvere-Tierney topology that expresses the notion of "mu-almost everywhere equivalence" induces a subtopos Shmu( L ). If this subtopos is Boolean, and if mu is locally finite, then the Radon-Nikodym theorem holds, so that for any locally finite nu << mu, the Radon-Nikodym derivative dndm exists. |
