
theorem Th49:
  for X be non empty set, S be SigmaField of X, M be sigma_Measure
of S, f be PartFunc of X,ExtREAL st ( ex A be Element of S st dom f = A ) for c
be Real, B be Element of S st f is B-measurable holds c(#)f is B-measurable
proof
  let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be
  PartFunc of X,ExtREAL;
  assume ex A be Element of S st A = dom f;
  then consider A be Element of S such that
A1: A = dom f;
  let c be Real, B be Element of S;
  assume f is B-measurable;
  then f is (A/\B)-measurable by A1,Th48;
  then
A2: c(#)f is (A/\B)-measurable by A1,MESFUNC1:37,XBOOLE_1:17;
  dom(c(#)f) = A by A1,MESFUNC1:def 6;
  hence thesis by A2,Th48;
end;
