
theorem Th87:
  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 A = dom f & f
is A-measurable) & (for x be Element of X st x in dom f holds 0= f.x) holds
  integral+(M,f) = 0
proof
  let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be
  PartFunc of X,ExtREAL such that
A1: ex A be Element of S st A = dom f & f is A-measurable and
A2: for x be Element of X st x in dom f holds 0 = f.x;
A3: for x be object st x in dom f holds 0 <= f.x by A2;
A4: dom(0(#)f) =dom f by MESFUNC1:def 6;
  now
    let x be object;
    assume
A5: x in dom(0(#)f);
    hence (0(#)f).x = 0 * f.x by MESFUNC1:def 6
      .= 0
      .= f.x by A2,A4,A5;
  end;
  hence integral+(M,f) = integral+(M,0(#)f) by A4,FUNCT_1:2
    .= 0 * integral+(M,f) by A1,A3,Th86,SUPINF_2:52
    .= 0;
end;
