
theorem Th34:
for a,b be Real, f be PartFunc of REAL,REAL st
].a,b.] c= dom f &
 f is_left_improper_integrable_on a,b
 holds for E be Element of L-Field st E c= ].a,b.] holds f is E-measurable
proof
    let a,b be Real, f be PartFunc of REAL,REAL;
    assume that
A1: ].a,b.] c= dom f and
A2:  f is_left_improper_integrable_on a,b;

    per cases;
    suppose A3: a < b; then
    reconsider A = ].a,b.] as non empty Subset of REAL by XXREAL_1:32;
    reconsider A as Element of L-Field by MEASUR12:72,75;

    set dif = b-a;

    consider K be SetSequence of L-Field such that
A4:  (for n be Nat holds K.n = [. a+(b-a)/(n+1), b .] & K.n c= ].a,b.] &
       K.n is non empty closed_interval Subset of REAL) &
     K is non-descending & K is convergent & Union K = ]. a,b .] by A3,Th24;
    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;

A5:  a< a+dif/(n+1) <= b by A3,Th22;
     reconsider Kn = K.n as non empty closed_interval Subset of REAL by A4;
     Kn = [.a+dif/(n+1),b.] by A4; then
     Kn = [' a+dif/(n+1),b '] by A5,INTEGRA5:def 3; then
A6:  f is_integrable_on (Kn) & f||Kn is bounded by A5,A2,INTEGR24:def 1;

     Kn c= A by A4; then
     Kn c= dom f by A1;
     hence (R_EAL f) is (K1.n)-measurable by A6,MESFUN14:49,MESFUNC6:def 1;
    end; then
    f is (Union K1)-measurable by Th21,MESFUNC6:def 1;
    hence thesis by A4,MESFUNC6:16;
    end;
    suppose a >= b; then
A7:  ].a,b.] = {} by XXREAL_1:26;
     hereby let E be Element of L-Field;
      assume E c= ].a,b.]; then
      E = {} by A7;
      hence f is E-measurable by Th30;
     end;
    end;
end;
