reserve X for non empty set,
        x for Element of X,
        S for SigmaField of X,
        M for sigma_Measure of S,
        f,g,f1,g1 for PartFunc of X,REAL,
        l,m,n,n1,n2 for Nat,
        a,b,c for Real;

theorem Th2:
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
A be Element of S, f be PartFunc of X,ExtREAL st
  A = dom f & f is A-measurable & f is nonnegative holds
   Integral(M,f) in REAL iff f is_integrable_on M
proof
   let X be non empty set,
       S be SigmaField of X,
       M be sigma_Measure of S,
       A be Element of S,
       f be PartFunc of X,ExtREAL;
   assume A1: A = dom f & f is A-measurable & f is nonnegative;
A2:now assume f is_integrable_on M; then
    -infty < Integral(M,f) & Integral(M,f) < +infty by MESFUNC5:96;
    hence Integral(M,f) in REAL by XXREAL_0:14;
   end;
   now assume A3: Integral(M,f) in REAL;
A4: dom (max-f) = A & max-f is A-measurable by A1,MESFUNC2:26,def 3;
A5: dom (max+f) = A & max+f is A-measurable by A1,MESFUNC2:25,def 2;
    for x being Element of X holds 0 <= (max+f).x by MESFUNC2:12; then
    max+f is nonnegative by SUPINF_2:39; then
A6: Integral(M,max+f) = integral+(M,max+f) by A5,MESFUNC5:88;
A7:for x be Element of X st x in dom f holds (max+f).x = f.x
    proof
     let x be Element of X;
A8:  f.x >= 0 by A1,SUPINF_2:39;
     assume x in dom f; then
     (max+f).x = max(f.x,0) by A1,A5,MESFUNC2:def 2;
     hence thesis by A8,XXREAL_0:def 10;
    end; then
    max+f = f by A1,A5,PARTFUN1:5; then
A9: Integral(M,max+f) < +infty by A3,XXREAL_0:9;
    for x being Element of X holds 0 <= (max-f).x by MESFUNC2:13; then
    max-f is nonnegative by SUPINF_2:39; then
A10: Integral(M,max-f) = integral+(M,max-f) by A4,MESFUNC5:88;
    for x be Element of X st x in dom (max-f) holds 0= (max-f).x
    proof
     let x be Element of X;
     assume x in dom (max-f);
     max+f.x = f.x by A1,A5,A7,PARTFUN1:5;
     hence 0= max-f.x by MESFUNC2:19;
    end; then
    Integral(M,max-f) = 0 by A4,LPSPACE1:22;
    hence f is_integrable_on M by A1,A6,A9,A10;
   end;
   hence thesis by A2;
end;
