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Jaynes In his paper "Information theory and Statistical mechanics" says

"Previously, one constructed a theory based on equations of motion, supplemented by additional hypothesis of ergodicity, Metric transitivity, or equal a priori probabilities, and the identification of entropy was made only at the end, by comparison of the resulting equation with the laws of phenomenological thermodynamics."

In this context what does "Metric Transitivity" mean?

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It is explained on the first page of
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1076959/pdf/pnas01784-0023.pdf

''If E is any invariant measurable set, then either E or its complement is of measure zero.'''

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  • $\begingroup$ I still don't understand (I am not a mathematician). How would it "translate" in the material/physical world? Could you give a concrete example and a counter-example? $\endgroup$
    – The Quark
    Apr 30, 2023 at 13:05
  • $\begingroup$ @TheQuark: The invariant sets are the orbits. The orbit of a point is the set of elements to which the point can be moved by an element of the group of transformations under consideration. The statement is equivalent that there is only a single orbit with positive measure (and hence nonzero probability). $\endgroup$
    – Arnold Neumaier
    May 1, 2023 at 14:58

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