
theorem Th67:
for X be non empty set, F be Functional_Sequence of X,ExtREAL,
 S be SigmaField of X, E be Element of S, m be Nat st
  F is with_the_same_dom & E = dom(F.0) &
  (for n be Nat holds F.n is E-measurable & F.n is without+infty)
  holds (Partial_Sums F).m is E-measurable
proof
    let X be non empty set, F be Functional_Sequence of X,ExtREAL,
    S be SigmaField of X, E be Element of S, m be Nat;
    assume that
A1:  F is with_the_same_dom and
A2:  E = dom(F.0) and
A3:  for n be Nat holds F.n is E-measurable & F.n is without+infty;
    now let n be Nat;
     E = dom(F.n) by A1,A2,MESFUNC8:def 2; then
     -(F.n) is E-measurable by A3,MEASUR11:63;
     hence (-F).n is E-measurable by Th37;
     F.n is without+infty by A3; then
     -(F.n) is without-infty;
     hence (-F).n is without-infty by Th37;
    end; then
    (Partial_Sums (-F)).m is E-measurable by MESFUNC9:41; then
    (-(Partial_Sums F)).m is E-measurable by Th42; then
A5: -((Partial_Sums F).m) is E-measurable by Th37;
    dom((Partial_Sums F).m) = E by A1,A2,A3,Th46,MESFUNC9:29; then
    dom(-((Partial_Sums F).m)) = E by MESFUNC1:def 7; then
    -(-((Partial_Sums F).m)) is E-measurable by A5,MEASUR11:63;
    hence (Partial_Sums F).m is E-measurable by DBLSEQ_3:2;
end;
