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uncountable sums of positive numbers
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(Definition)
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The notion of sum of a series can be generalized to sums of nonnegative real numbers over arbitrary index sets.
let  be a set and let  be a mapping from  to the nonnegative real numbers. Then we may define the sum as follows:
In words, we are taking the supremum over all sums over finite subsets of the index set. This agrees with the usual notion of sum when our set is countably infinite, but generalizes this notion to uncountable index sets.
An important fact about this generalization is that the sum can only be finite if the number of elements  such that  is countable. To demonstrate this fact, define the sets  (where n is a nonnegative integer) as follows:
when  ,
If any of these sets is infinite, then the sum will diverge so, for the sum to be finite, all these sets must be finite. However, if these sets are all finite, then their union is countable. In other words, the number of indices for which  will be countable.
This notion finds use in places such as non-separable Hilbert spaces. For instance, given a vector in such a space and a complete orthonormal set, one can express the norm of the vector as the sum of the squares of its components using this definition even when the orthonormal set is uncountably infinite.
This discussion can also be phrased in terms of Lesbegue integration with respect to counting measure. For this point of view, please see the entry support of integrable function with respect to counting measure is countable.
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"uncountable sums of positive numbers" is owned by rspuzio.
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(view preamble)
Cross-references: support of integrable function with respect to counting measure is countable, point, counting measure, terms, even, components, squares, norm, orthonormal set, complete, vector, Hilbert spaces, places, indices, union, diverge, infinite, integer, countable, number, uncountable, countably infinite, subsets, finite, supremum, mapping, index sets, real numbers, series, sum
This is version 2 of uncountable sums of positive numbers, born on 2006-03-08, modified 2006-03-08.
Object id is 7698, canonical name is UncountableSumsOfPositiveNumbers.
Accessed 1418 times total.
Classification:
| AMS MSC: | 40-00 (Sequences, series, summability :: General reference works ) |
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Pending Errata and Addenda
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