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The first section consists of auxiliary results about nondecreasing real functions. In the second section a new characterization of relatively compact sets of regulated functions in the sup-norm topology is brought, and the third section includes, among others, an analogue of Helly's Choice Theorem in the space of regulated functions.

Assume that for any $t$ from an interval $[a,b]$ a real number $u\left(t\right)$ is given. Summarizing all these numbers $u\left(t\right)$ is no problem in case of an absolutely convergent series ${\sum }_{t\in [a,b]}u\left(t\right)$. The paper gives a rule how to summarize a series of this type which is not absolutely convergent, using a theory of generalized Perron (or Kurzweil) integral.

The generalized linear differential equation $dx=d\left[a\right(t\left)\right]x+df$ where $A,f\in B{V}_n^{loc}\left(J\right)$ and the matrices $I-{\Delta }^{-}A\left(t\right),I+{\Delta }^{+}A\left(t\right)$ are regular, can be transformed $\frac{dy}{ds}=B\left(s\right)y+g\left(s\right)$ using the notion of a logarithimc prolongation along an increasing function. This method enables to derive various results about generalized LDE from the well-known properties of ordinary LDE. As an example, the variational stability of the generalized LDE is investigated.

This paper deals with regulated functions having values in a Banach space. In particular, families of equiregulated functions are considered and criteria for relative compactness in the space of regulated functions are given.