Nonabsolutely convergent integrals

Title: Nonabsolutely convergent integrals Author: Krist'yna Kuncov'a Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Jan Mal'y, DrSc., Department of Mathematical Analysis Abstract: In this thesis we develop the theory of nonabsolutely convergent Hen- stock-Kurzweil type packing integrals in different spaces. In the framework of metric spaces we define the packing integral and the uniformly controlled inte- gral of a function with respect to metric distributions. Applying the theory to the notion of currents we then prove a generalization of the Stokes theorem. In Rn we introduce the packing R and R∗ integrals, which are defined as charges - additive functionals on sets of bounded variation. We provide comparison with miscellaneous types of integrals such as R and R∗ integral in Rn or MCα integral in R. On the real line we then study a scale of integrals based on the so called p-oscillation. We show that our indefinite integrals are a.e. approximately differ- entiable and we give comparison with other nonabsolutely convergent integrals. Keywords: Nonabsolutely convergent integrals, BV sets, Henstock-Kurzweil in- tegral, Divergence theorem, Analysis in metric measure spaces