reserve a,b,r for Real;
reserve A,B for non empty set;
reserve f,g,h for Element of PFuncs(A,REAL);
reserve u,v,w for VECTOR of RLSp_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 for Element of S,
  f,g,h,f1,g1 for PartFunc of X ,REAL;
reserve v,u for VECTOR of RLSp_L1Funct M;
reserve v,u for VECTOR of RLSp_AlmostZeroFunct M;

theorem Th44:
  f is_integrable_on M implies Integral(M,f) in REAL & Integral(M,
  abs f) in REAL & abs f is_integrable_on M
proof
  assume
A1: f is_integrable_on M;
  then
A2: -infty < Integral(M,f) & Integral(M,f) < +infty by MESFUNC6:90;
  R_EAL f is_integrable_on M by A1;
  then consider A be Element of S such that
A3: A = dom R_EAL f and
A4: R_EAL f is A-measurable;
A5: f is A-measurable by A4;
  then abs f is_integrable_on M by A1,A3,MESFUNC6:94;
  then -infty < Integral(M,abs f) & Integral(M,abs f) < +infty by MESFUNC6:90;
  hence thesis by A1,A2,A3,A5,MESFUNC6:94,XXREAL_0:14;
end;
