
theorem Th29:
for f,g be PartFunc of REAL,REAL st f a.e.= g,B-Meas holds f a.e.= g,L-Meas
proof
    let f,g be PartFunc of REAL,REAL;
    assume f a.e.= g,B-Meas; then
    consider E be Element of Borel_Sets such that
A1:  B-Meas.E = 0 & f|E` = g|E` by LPSPACE1:def 10;

    {} is empty & {} c= REAL; then
    reconsider E0 = {} as Element of Borel_Sets by MEASUR12:72;

A2: E = E \/ E0; then
    reconsider E1 = E as Element of L-Field
      by MEASURE3:def 3,MEASUR12:73,def 11;

    (COM B-Meas).E1 = 0 by A1,A2,MEASURE3:def 5,MEASUR12:73;
    hence f a.e.= g,L-Meas by A1,LPSPACE1:def 10,MEASUR12:def 12;
end;
