Compactness in the spaces of functions of bounded variation

Recently the characterization of the compactness in the space $BV([0,1])$ of functions of bounded Jordan variation was given. Here, certain generalizations of this result are given for the spaces of functions of bounded Waterman $\mathrm{\Lambda }$-variation, Young $\mathrm{\Phi }$-variation and integral variation. It appears that on the compact sets the norm is uniformly approximated by certain seminorms induced by the selection of finitely many intervals in $[0,1]$.