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Suppose $\mathcal{I}$ is an arbitrary indexing set, and suppose we look at a family $A_i$ of subsets of some set $E$. Let's say we fix an element $x \in E$ and we define the function $f: \mathcal{I} \to \{0,1\}$ by \begin{equation} f_x(i) = \begin{cases} 1 &\text{if } x \in A_i\\ 0 &\text{otherwise} \end{cases} \end{equation}

Now, does the function $F: E \to \mathbb{R} \cup \{\infty\}$ given by \begin{equation} F(x) = \sum_{i \in I} f_x(i) \end{equation} make sense?

I am asking this because I came upon a comment that says the summation symbol $\sum$ cannot be used in cases where the indexing set $\mathcal{I}$ is uncountable, instead, on has to use integration theory for this (and therefore, the symbol $\int$).

Yet to me it doesn't look as if the above function $F$ is ill - defined. What am I missing?

Thanks for your feedback!

  • 0
    it will not be very interesting. Summing an uncountable set of positive numbers will always result in infinity...2012-02-05
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    The way it is defined, it does make sense as a map from $E$ into $\mathbb{R}\cup \left\{\infty\right\}$, when $\infty$ is used as a formal symbol to mean the value of $f$ whenever for infinitely (countably or uncountably) many $i\in I$, $f_x(i) = 1$. I don't see a problem.2012-02-05

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