A Measure-theoretical dynamical system is comprised of:

• a Measurable space
• a measurable map
• a non-singular Measure

This space can be considered, without restriction to be a Probability space. See Amigo's book.

See here (local).

A Dynamical system on a Measurable space has a natural or physical invariant measure, corresponding to the Probability measure that numerical simulations of the system would produce asymptotically.