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-Nikodým 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 Kristýna Kuncová and Petr Honzík.
