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Uncountable Sum

  1. Aug 3, 2009 #1
    Does there exist a converging uncountable sum of strictly positive reals?
     
  2. jcsd
  3. Aug 4, 2009 #2
    Would that be an integral?
     
  4. Aug 4, 2009 #3
    No. I mean an actual uncountable sum. An (Riemann) integral is the limit of a sequence of countable sums.
     
  5. Aug 4, 2009 #4
    In the sensible way to define uncountable sums, you prove that for a sum of real terms, if it converges (to a real number) then all but countably many terms must be zero.
     
  6. Aug 4, 2009 #5

    HallsofIvy

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    First you will have to define what you mean by "uncountable sum"! I know a definition for finite sums and I know a definition for countable sums (the limit of the partial, finite, sums), but I do not know any definition for an uncountable sum except, possibly the integral that bpet suggested.
     
  7. Aug 4, 2009 #6
    Definition Let [itex]S[/itex] be an index set. Let [itex]a \colon S \to \mathbb{R}[/itex] be a real function on [itex]S[/itex]. Let [itex]V[/itex] be a real number. Then we say [itex]V = \sum_{s\in S} a(s)[/itex] iff for every [itex]\epsilon > 0[/itex] there is a finite set [itex]A_\epsilon \subseteq S[/itex] such that for all finite sets [itex]A[/itex] , if [itex]A_\epsilon \subseteq A \subseteq S[/itex] we have [itex]\left|V - \sum_{s \in A} a(s)\right| < \epsilon[/itex] .
     
  8. Aug 4, 2009 #7
    I'm surprised. I thought this would have been defined at some point.

    Suppose [tex]x_i[/tex] is a (possibly uncountable) sequence of positive reals indexed by some ordinal L. Then their sum is [tex]\sum_{i\in L}x_i=\sup\{\sum_{i\in k}x_i:k<L\}[/tex]. This takes care of limit ordinals.

    So does there exist sequences [tex]x_i[/tex] indexed by ordinals [tex]D\geq\epsilon_0[/tex] such that [tex]\sum_{i\in D}x_i[/tex] is finite, and that each x_i is positive?
     
  9. Aug 4, 2009 #8
    I don't know what this definition is trying to achieve. I prefer mine.
     
  10. Aug 5, 2009 #9
    If and only if [itex]D[/itex] is countable.
     
  11. Aug 5, 2009 #10
    Well [tex]\epsilon_0[/tex] is the first uncountable ordinal, so why not?
     
  12. Aug 5, 2009 #11
    Strange ... [itex]\epsilon_0[/itex] is commonly used to represent a certain countable ordinal, while [itex]\omega_1[/itex] denotes the least uncountable ordinal. In any case, that notation doesn't matter. Here is a repeat of the same answer as before: If a sum of positive real terms converges to a finite value, then the index set is countable.
     
  13. Aug 5, 2009 #12
    Yes I was mistaken on the notation, it should be [tex]\omega_1[/tex].

    You have yet to say why. You asserting it true doesn't constitute a proof.
     
  14. Aug 6, 2009 #13
    Hint for the proof: the real line has a countable dense set, and every term of the convergent series corresponds to an interval.
     
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