  For any real-valued stochastic process X with c\`adl\`ag paths we define
non-empty family of processes, which have finite total variation, have jumps of
the same order as the process X and uniformly approximate its paths: This
allows to decompose any real-valued stochastic process with c\`adl\`ag paths
and infinite total variation into a sum of uniformly close, finite variation
process and an adapted process, with arbitrary small amplitude but infinite
total variation. Another application of the defined class is the definition of
the stochastic integral with respect to the process X as a limit of pathwise
Lebesgue-Stieltjes integrals. This construction leads to the stochastic
integral with some correction term.
