reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S,
  F,G for Functional_Sequence of X,ExtREAL,
  I for ExtREAL_sequence,
  f,g for PartFunc of X,ExtREAL,
  seq, seq1, seq2 for ExtREAL_sequence,
  p for ExtReal,
  n,m for Nat,
  x for Element of X,
  z,D for set;

theorem Th44:
  dom(F.0) = E & F is additive & F is with_the_same_dom & (for n
be Nat holds (Partial_Sums F).n is E-measurable) & (for x be Element of X 
st
  x in E holds F#x is summable) implies lim(Partial_Sums F) is E-measurable
proof
  assume that
A1: dom(F.0) = E and
A2: F is additive and
A3: F is with_the_same_dom and
A4: for n be Nat holds (Partial_Sums F).n is E-measurable and
A5: for x be Element of X st x in E holds F#x is summable;
  reconsider PF = Partial_Sums F as with_the_same_dom Functional_Sequence of X
  ,ExtREAL by A2,A3,Th43;
A6: now
    let x be Element of X;
    assume
A7: x in E;
    then F#x is summable by A5;
    then Partial_Sums(F#x) is convergent;
    hence PF#x is convergent by A1,A2,A3,A7,Th33;
  end;
  dom((Partial_Sums F).0) = E by A1,A2,A3,Th29;
  hence thesis by A4,A6,MESFUNC8:25;
end;
