A proof of two conjectures related to the Erdős-Debrunner inequality.
In this paper, we define a -integral, i.e. an integral defined by means of partitions of unity, on a suitable compact metric measure space, whose measure is compatible with its topology in the sense that every open set is -measurable. We prove that the -integral is equivalent to -integral. Moreover, we give an example of a noneuclidean compact metric space such that the above results are true.
This note contains a proof of the existence of a one-to-one function of onto itself with the following properties: is a rational-linear automorphism of , and the graph of is a non-measurable subset of the plane.
An exact Radon-Nikodym derivative is obtained for a pair (I,J) of positive linear functionals, with J absolutely continuous with respect to I, using a notion of exhaustion of I on elements of a function algebra lattice.
We prove a relaxation theorem in BV for a non coercive functional with linear growth. No continuity of the integrand with respect to the spatial variable is assumed.