definition of sum of uncountable number of non-negative real numbers
Topics
About: metric space
The table of contents of this article
Starting Context
- The reader knows a definition of Euclidean metric space.
- The reader knows a definition of convergence of sequence on metric space.
- The reader knows a definition of supremum of subset of partially-ordered set.
Target Context
- The reader will have a definition of sum of uncountable number of non-negative real numbers.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( J\): \(\in \{\text{ the uncountable index sets }\}\)
\( \mathbb{R}\): \(= \text{ the Euclidean metric space }\) with the canonical linear ordering
\( \{r_j \in [0, \infty) \subseteq \mathbb{R} \vert j \in J\}\):
\(*\sum_{j \in J} r_j\): \(= Sup (\{\sum_{j \in J^`} r_j \vert J^` \in \{\text{ the countable subsets of } J\}\}) \text{ when it exists }; \infty \text{ when it does not exist }\)
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Conditions:
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2: Note
The reason why we need to introduce countable subsets is that we cannot take the sum of any uncountable number of numbers directly: \(\sum_{j \in J^`} r_j\) is taken as the convergence of a sequence (although the order of \(J^`\) is not specified, the result does not depend on the order as the terms are non-negative, as is well known).
The reason why we think of only the sums of non-negative numbers is that otherwise, \(\sum_{j \in J^`} r_j\) would depend on the order of \(J^`\) (possibly undefined with the sequence not converging) and taking the supremum of such results does not seem so meaningful.
