
theorem Th38:
for f be PartFunc of REAL,REAL st dom f = REAL
 & f is_improper_integrable_on_REAL holds
 for E be Element of L-Field holds f is E-measurable
proof
    let f be PartFunc of REAL,REAL;
    assume that
A1:  dom f = REAL and
A2:  f is_improper_integrable_on_REAL;

    reconsider A = [.0,+infty.[, B = ].-infty,0 .] as Element of L-Field
      by MEASUR10:5,MEASUR12:75;
A3: A \/ B = REAL by XXREAL_1:172,224;

    f is_-infty_improper_integrable_on 0
  & f is_+infty_improper_integrable_on 0 by A1,A2,INTEGR25:36; then
    (R_EAL f) is A-measurable & (R_EAL f) is B-measurable
      by A1,Th36,Th37,MESFUNC6:def 1; then
    (R_EAL f) is (A \/ B)-measurable by MESFUNC1:31;
    hence thesis by A3,MESFUNC6:def 1,16;
end;
