Winter School

Nonabsolutely convergent integrals with respect to distributions

We introduce integrals of functions with respect to distributions. This integration is an inverse process to a Radon-Nikodm type differentiation of distributions with respect to distributions. We can develop this integration also in the setting of metric distributions and currents on metric spaces. The new integral is non-absolutely convergent, similarly to the Denjoy-Perron (or Henstock-Kurzweil) integral. The process of integration is new even for integration with respect to the Lebesgue measure on the real line. With the aid of this integral, we can study structures resembling currents or varifolds and establish some generalizations of the Stokes theorem. Also, using the new integration process we can introduce non-absolutely convergent singular integrals. Our view point gives an alternative to other methods like taking principal values. The presented results are a joint work with Kristna Kuncov and Petr Honzk.