Countably modulared spaces of functions of bounded variation (Q1089572)

In the theory of modular spaces there are no general conditions for the equalities (or inclusions) of several countably modulared spaces to hold. The purpose of this paper is to investigate the problem of equality of the countable modulared spaces \(X_{\rho},X_{\rho_ 0},X_{\rho_ o}\), and \(X_{\rho_ s}\) with modulars defined by \(M_ i\)-variations \((M_ i:\) \(R\to R\) are continuous functions non-negative and non- decreasing in \(R^+\) and such that \(M_ i(u)=0\) iff \(u=0)\). The inclusions between \(V_ 1\) and \(X_{\rho_ s}\) and between \(V_ 1\) and \(X_{\rho_ 0}\) are also described \((V_ 1\) is the class of all functions of bounded variation in the usual sense, i.e. with M(u)\(\equiv | u|)\). Some examples of sequences \(\{M_ i\}\) for which the above equalities and inclusions hold are given.