Remark on BV-solutions of a functional equation connected with invariant measures (Q1070129)

The functional equation of the type \[ (1)\quad \Phi (x)=\sum \{g_ i\Phi (f_ i(x)):\quad i=1,\ldots,n\},\quad x\in [0,1], \] appears e.g. when dealing with the question of invariant Borel measures for piecewise monotonic mappings \(T: [0,1]\to [0,1].\) The author shows that under some general assumptions a function of bounded variation satisfies (1) if and only if its upper and lower variations do. Thus all BV-solutions of (1) will be known once all the monotonic ones are determined.