We begin with the celebrated result of [3]. The authors were well aware that their result cannot be extended to expanding transformations with countably many one-to-one pieces in a simple way (see Th. 2, and the comment below on Cond. (17) there). The real task in that period of time was to find reasonable additional conditions which would guarantee the existence of density invariant under the action of expanding map with countably many one-to-one pieces. Several attempts was made to accomplish that task (for more details see e.g. a review article [4], and also [5], or [6], Sect. 6).

One of the mentioned attempts was published in [7], as Adler’s Theorem. Since no proof was given there, the question arose whether it is true [8]. A solution was published in [1], and [2].

After the above two notes and a few other ones, related with them, were published, some further claims and opinions concerning the existence of invariant densities and their lower and upper bounds for Markov Maps appear in the literature.

Those claims and opinions reveal that their authors were unacquainted with the essence of the problem. That problem is rather of delicate nature. It involves, among other things, the so-called measure-theoretic recurrence property.

In this note we clear up, in a systematic way, the essence of the problems with the aid of examples, comments and some published results.