Observers and relative entropy of G-sets

In the current study, a new type of relative entropy is presented through observable objects in quantum mechanics as the probability measures. The relative probability measures as an extension of probability measures are considered via one-dimensional observers. The notion of relative entropy of the semigroup on a relative measure space is introduced using the mathematical modeling of an observable object. Moreover, it is proved that this nonnegative quantity is invariant under $({\Theta }_1,{\Theta }_2)$-isomorphism. Finally, we applied this concept to specific semigroups to extract Kolmogorov entropy of a dynamical system as a special case.