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A measure-preserving system in which every invariant measurable set has measure zero or full measure. Ergodic theorems then relate long-time averages to conditional or space averages under their stated integrability assumptions.
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This is an authored directed relation from the source endpoint to the target endpoint.
Ergodicity imposes that every invariant measurable set has measure zero or full measure; measure preservation by itself is weaker.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
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This is an authored directed relation from the source endpoint to the target endpoint.
Measure-theoretic ergodicity supplies the precise invariant-set condition invoked by the physical ergodic hypothesis; actual equilibrium statistical mechanics does not require every microscopic system to be strictly ergodic.
A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.
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