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;

theorem Th24:
  f in L1_Functions M implies a(#)f in L1_Functions M
proof
  set W = L1_Functions M;
  assume f in W;
  then
  ex f1 be PartFunc of X,REAL st f1=f & ex ND be Element of S st M.ND=0 &
  dom f1 = ND` & f1 is_integrable_on M;
  then consider ND be Element of S such that
A1: M.ND=0 and
A2: dom f = ND` & f is_integrable_on M;
  dom (a(#)f) = ND` & a(#)f is_integrable_on M by A2,MESFUNC6:102
,VALUED_1:def 5;
  hence thesis by A1;
end;
