Multi-dimensional observers and relative entropy of dynamical systems

In this paper the notion of relative probability‎ ‎measure of a set E is considered with respect to a multi-dimensional observer of a set‎ X as a superset of E‎. ‎Relative entropy of a multi-dimensional‎ ‎observer for the partitions is defined and the properties of relative entropy is extended to‎ ‎multi-dimensional observers‎. ‎It is shown that the observer of a set plays a‎ ‎role in uncertainty of a partition of it‎. ‎Relative conditional entropy is also considered and its main properties are proved‎. ‎Moreover‎, ‎the relative entropy off‎ ‎a relative measure preserving map is studied as well‎.