 reserve a,b,r for Complex;
 reserve V for ComplexLinearSpace;
reserve A,B for non empty set;
reserve f,g,h for Element of PFuncs(A,COMPLEX);
reserve u,v,w for VECTOR of CLSp_PFunctA;
reserve X for non empty set,
  x for Element of X,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E,E1,E2,A,B for Element of S,
  f,g,h,f1,g1 for PartFunc of X,COMPLEX;
reserve v,u for VECTOR of CLSp_L1Funct M;
reserve v,u for VECTOR of CLSp_AlmostZeroFunct M;

theorem Th37:
  f is_integrable_on M implies Integral(M,f) in COMPLEX &
  Integral(M,|.f.|) in REAL & |.f.| is_integrable_on M
proof
  assume
A1: f is_integrable_on M;
  reconsider AF = |.f.| as PartFunc of X,REAL;
  AF is_integrable_on M by A1,MESFUN7C:35;
  hence thesis by LPSPACE1:44,XCMPLX_0:def 2;
end;
