
theorem Th50:
for I,J be non empty closed_interval Subset of REAL,
 f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
 g be PartFunc of [:REAL,REAL:],REAL,
 E be Element of sigma measurable_rectangles(L-Field,L-Field)
 st [:I,J:] = dom f & f is_continuous_on [:I,J:] & f = g & E = [:I,J:] holds
   g is E-measurable
proof
    let I,J be non empty closed_interval Subset of REAL,
    f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
    g be PartFunc of [:REAL,REAL:],REAL,
    E be Element of sigma measurable_rectangles(L-Field,L-Field);
    assume that
A1: [:I,J:] = dom f and
A2: f is_continuous_on [:I,J:] and
A3: f = g and
A4: E = [:I,J:];

    for r being Real holds E /\ (less_dom (g,r))
     in sigma measurable_rectangles(L-Field,L-Field)
    proof
     let r be Real;
A5:  less_dom(g,r) = g" ( ].-infty,r.[ ) by Th17;
     consider H be Subset of [:RNS_Real,RNS_Real:] such that
A6:   H /\ E = f" ( ].-infty,r.[ ) & H is open by Th24,A1,A2,A4;
     H in sigma measurable_rectangles(L-Field,L-Field) by A6,Th15; then
     f" ( ].-infty,r.[ ) in sigma measurable_rectangles(L-Field,L-Field)
      by A6,FINSUB_1:def 2;
     hence E /\ (less_dom (g,r)) in
      sigma measurable_rectangles(L-Field,L-Field) by A3,A5,FINSUB_1:def 2;
    end;
    hence g is E-measurable by MESFUNC6:12;
end;
