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The evolution of the phase space density of particle beams in external fieldsis found proceeding from the continuity equation in the six-dimensional (6D)phase space (mu-space). The Robinson theorem, which includes the Liouvilletheorem as a special case, was proved in a more simple and consistentalternative way valid for arbitrary external fields, averaged fields of thebeam (self-generated electro-magnetic fields except intrabeam scattering) andarbitrary frictional forces (linear, nonlinear). It includes particleaccelerators as a special case. The limits of the applicability of the Robinsontheorem in case of cooling of excited ions having a finite living time arepresented.