
theorem Th36:
for a be Real, f be PartFunc of REAL,REAL
 st [.a,+infty.[ c= dom f & f is_+infty_improper_integrable_on a holds
 for E be Element of L-Field st E c= [.a,+infty.[ holds f is E-measurable
proof
    let a be Real, f be PartFunc of REAL,REAL;
    assume that
A1:  [.a,+infty.[ c= dom f and
A2:  f is_+infty_improper_integrable_on a;

    set A = [.a,+infty.[;
    reconsider A as Element of L-Field by MEASUR12:72,75;

    consider K be SetSequence of L-Field such that
A3:  (for n be Nat holds K.n = [.a,a+n.]) &
     K is non-descending & K is convergent & Union K = [.a,+infty.[ by Th25;

    rng K c= L-Field; then
    reconsider K1 = K as sequence of L-Field by FUNCT_2:6;

    for n be Nat holds (R_EAL f) is (K1.n)-measurable
    proof
     let n be Nat;
A4:  a<= a+n by XREAL_1:31;

A5:  K.n = [.a,a+n.] by A3; then
     reconsider Kn = K.n as non empty closed_interval Subset of REAL
       by XREAL_1:31,XXREAL_1:30,MEASURE5:def 3;
     Kn = [' a,a+n '] by A5,XREAL_1:31,INTEGRA5:def 3; then
A6:  f is_integrable_on Kn & f||Kn is bounded by A4,A2,INTEGR25:def 2;

     Kn c= A by A5,XXREAL_1:251; then
     Kn c= dom f by A1;
     hence (R_EAL f) is (K1.n)-measurable by A6,MESFUN14:49,MESFUNC6:def 1;
    end; then
    (R_EAL f) is A-measurable by A3,Th21;
    hence thesis by MESFUNC6:def 1,16;
end;
