On interplay between jet and information geometries

We will consider the procedure of measurement of random vectors, operators and tensors from the double point of view: pure probabilistic and geometrical. Using the principle of minimum information gain, we reformulate the probabilistic approach as studies in the geometry of jet spaces over the manifolds of extreme measures. Moreover, the procedure of a measurement itself becomes equivalent to study various geometrical structures on integral manifolds of the Cartan distribution. We will illustrate all of this for the case of thermodynamics of real gases and phase transitions of the first and second orders.