
theorem Th35:
for a,b be Real, f be PartFunc of REAL,REAL st ].a,b.[ c= dom f &
 f is_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_improper_integrable_on a,b;

    consider c be Real such that
A3:  a < c < b and
A4:  f is_left_improper_integrable_on a,c and
A5:  f is_right_improper_integrable_on c,b and
     not (left_improper_integral(f,a,c) = -infty
        & right_improper_integral(f,c,b) = +infty)
   & not (left_improper_integral(f,a,c) = +infty
        & right_improper_integral(f,c,b) = -infty) by A2,INTEGR24:def 5;

    hereby let E be Element of L-Field;
     assume A6: E c= ].a,b.[;
     ].a,c.] c= ].a,b.[ by A3,XXREAL_1:49; then
A7:  ].a,c.] c= dom f by A1;
     [.c,b.[ c= ].a,b.[ by A3,XXREAL_1:48; then
A8:  [.c,b.[ c= dom f by A1;

     reconsider L = ].a,c.] as Element of L-Field by MEASUR10:5,MEASUR12:75;
A9:  f is (E/\L)-measurable by A7,A4,Th34,XBOOLE_1:17;
     reconsider R = [.c,b.[ as Element of L-Field by MEASUR10:5,MEASUR12:75;
A10:  L\/R = ].a,b.[ by A3,XXREAL_1:172;
A11:  E/\L \/ E/\R = E/\(L\/R) by XBOOLE_1:23
      .= E by A6,A10,XBOOLE_1:28;

     f is (E/\R)-measurable by A8,A5,Th33,XBOOLE_1:17;
     hence f is E-measurable by A11,A9,MESFUNC6:17;
    end;
end;
