reserve Omega for non empty set;
reserve r for Real;
reserve Sigma for SigmaField of Omega;
reserve P for Probability of Sigma;
reserve E for finite non empty set;
reserve f,g for Real-Valued-Random-Variable of Sigma;

theorem Th23:
  for r be Real st 0 <= r & f is nonnegative holds (f
  to_power r) is Real-Valued-Random-Variable of Sigma
proof
  let r be Real;
  assume
A1: 0 <= r & f is nonnegative;
A2: dom f = Omega by FUNCT_2:def 1;
  rng (f to_power r) c= REAL & dom (f to_power r) = dom f by MESFUN6C:def 4;
  then
A3: (f to_power r) is Function of Omega,REAL by A2,FUNCT_2:2;
  dom f = [#]Sigma by FUNCT_2:def 1; then
  (f to_power r) is ([#]Sigma)-measurable by A1,MESFUN6C:29;
  hence thesis by A3;
end;
