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.