In this paper we introduce a parameter in the space of functions $L_0$. The parameter measures the lack of equimeasurability using a sequence of functions which controls the oscillations of every function of a given subset of $L_0$. We estimate the Hausdorff measure of noncompactness in terms of and the parameter a (see [3]) and characterize the totally bounded subsets of $L_0$. A criterion of compactness given in [5] for subsets of the space $BC(\Omega,R)$ is extended to the case of the space $BTC(\Omega,M)$.