reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,COMPLEX,
  r for Real,
  c for Complex,
  E,A,B for Element of S;

theorem Th29:
  for k be Real, X,S,E for f be PartFunc of X,REAL st f is
  nonnegative & 0 <= k & E c= dom f & f is E-measurable holds (f to_power k)
  is E-measurable
proof
  let k be Real, X,S,E;
  let f be PartFunc of X,REAL;
  reconsider k1=k as Real;
  assume that
A1: f is nonnegative and
A2: 0 <= k and
A3: E c= dom f and
A4: f is E-measurable;
A5: dom(f to_power k) = dom f by Def4;
  per cases by A2;
  suppose
A6: k = 0;
A7: E c= dom (f to_power k) by A3,Def4;
    now
      let r be Real;
      reconsider r1=r as Real;
      per cases;
      suppose
A8:     r <= 1;
        now
          let x be object;
          assume
A9:       x in E;
          then (f to_power k).x = (f.x) to_power k by A3,A5,Def4;
          then r <= (f to_power k).x by A6,A8,POWER:24;
          hence x in great_eq_dom(f to_power k,r1) by A3,A5,A9,MESFUNC6:6;
        end;
        then E c= great_eq_dom(f to_power k,r) by TARSKI:def 3;
        then E /\ great_eq_dom(f to_power k,r) = E by XBOOLE_1:28;
        then E /\ less_dom(f to_power k,r1) = E \ E by A7,Th28;
        hence E /\ less_dom(f to_power k,r) in S;
      end;
      suppose
A10:    1 < r;
        now
          let x be object;
          assume
A11:      x in E;
          then (f to_power k).x = (f.x) to_power k by A3,A5,Def4;
          then (f to_power k).x < r by A6,A10,POWER:24;
          hence x in less_dom(f to_power k,r) by A3,A5,A11,MESFUNC6:3;
        end;
        then E c= less_dom(f to_power k,r) by TARSKI:def 3;
        then E /\ less_dom(f to_power k,r) = E by XBOOLE_1:28;
        hence E /\ less_dom(f to_power k,r) in S;
      end;
    end;
    hence thesis by MESFUNC6:12;
  end;
  suppose
A12: k > 0;
    for r be Real holds E /\ great_eq_dom(f to_power k,r) in S
    proof
      let r be Real;
      reconsider r1=r as Real;
      per cases;
      suppose
A13:    r1 <= 0;
        now
          let x be object;
          assume x in E;
          then x in dom f by A3;
          then
A14:      x in dom (f to_power k) by Def4;
          0 <= (f to_power k).x by A1,A12,MESFUNC6:51;
          hence x in great_eq_dom(f to_power k,r1) by A13,A14,MESFUNC6:6;
        end;
        then E c= great_eq_dom(f to_power k,r) by TARSKI:def 3;
        then E /\ great_eq_dom(f to_power k,r) = E by XBOOLE_1:28;
        hence thesis;
      end;
      suppose
A15:    0 < r1;
A16:    now
          set R = r to_power (jj/k);
          let x be object;
           reconsider xx=x as set by TARSKI:1;
A17:      0 < r to_power (jj/k) by A15,POWER:34;
          assume
A18:      x in great_eq_dom(f,r1 to_power (1/k));
          then
A19:      x in dom (f to_power k) by A5,MESFUNC6:6;
          R to_power k = r to_power (1/k*k) by A15,POWER:33;
          then
A20:      R to_power k = r to_power 1 by A12,XCMPLX_1:87;
          ex y be Real st y = f.x & r1 to_power (1/k) <= y
              by A18,MESFUNC6:6;
          then r1 to_power jj <= (f.xx) to_power k1
                 by A12,A17,A20,HOLDER_1:3;
          then r <= (f.xx) to_power k by POWER:25;
          then r <= (f to_power k).xx by A19,Def4;
          hence x in great_eq_dom(f to_power k,r1) by A19,MESFUNC6:6;
        end;
        now
          let x be object;
           reconsider xx=x as set by TARSKI:1;
          assume
A21:      x in great_eq_dom(f to_power k,r1);
          then
A22:      x in dom (f to_power k) by MESFUNC6:6;
          0 <= f.xx by A1,MESFUNC6:51;
          then
          ((f.xx) to_power k1) to_power (1/k1) = (f.xx) to_power (k1 * 1/k1
          ) by A12,HOLDER_1:2;
          then ((f.xx) to_power k1) to_power (1/k1) = (f.xx) to_power 1 by A12,
XCMPLX_1:87;
          then
A23:      ((f.xx) to_power k) to_power (1/k) = f.x by POWER:25;
          ex y2 be Real st y2 = (f to_power k).x & r1 <= y2
by A21,MESFUNC6:6;
          then r <= (f.xx) to_power k by A22,Def4;
          then
          r to_power (1/k1) <= ((f.xx) to_power k1) to_power (1/k1) by A12,A15,
HOLDER_1:3;
          hence x in great_eq_dom(f,r1 to_power (1/k)) by A5,A22,A23,MESFUNC6:6
;
        end;
        then E /\ great_eq_dom(f to_power k,r1) = E /\ great_eq_dom(f,r1
        to_power (1/k)) by A16,TARSKI:2;
        hence thesis by A3,A4,MESFUNC6:13;
      end;
    end;
    hence thesis by A3,A5,MESFUNC6:13;
  end;
end;
