  We present the development of the approach to thermodynamics based on
measurement. First of all, we recall that considering classical thermodynamics
as a theory of measurement of extensive variables one gets the description of
thermodynamic states as Legendrian or Lagrangian manifolds representing the
average of measurable quantities and extremal measures. Secondly, the variance
of random vectors induces the Riemannian structures on the corresponding
manifolds. Computing higher order central moments one drives to the
corresponding higher order structures, namely, the cubic and the fourth order
forms. The cubic form is responsible for the skewness of the extremal
distribution. The condition for it to be zero gives us so-called symmetric
processes. The positivity of the fourth order structure gives us an additional
requirement to thermodynamic state.
