A localic theory of lower and upper integrals

An account of lower and upper integration is given. It is constructive in the sense of geometric logic. If the integrand takes its values in the nonnegative lower reals, then its lower integral with respect to a valuation is a lower real. If the integrand takes its values in the non-negative upper reals, then its upper integral with respect to a covaluation and with domain of integration bounded by a compact subspace is an upper real. Spaces of valuations and of covaluations are defined. Riemann and Choquet integrals can be calculated in terms of these lower and upper integrals. This is a preprint version of the article published as – Mathematical Logic Quarterly 54 (1) (2008), pp. 109 123. doi:10.1002/malq.200710028