Introduction of the Kullback – Leibler information function by means of partitions of the probability space

In the paper we study the equivalence problem of two definitions of the Kullback – Leibler information function. It is commonly defined by integration of the logarithm of the density of one probability measure with respect to another. On the other hand, recently a concept of $t$-entropy of a dynamical system (that is a generalization of the information function) is actively explored, and this concept is defined by means of measurable partitions of the phase space. In the paper we investigate in which situations the two definitions are equivalent and in which ones they are not. In particular, the equivalence holds if both measures are finite.