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measure theory: show is countable

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Let {x_alpha}_[(alpha)(E)(gamma)] be an indexed collection of non-negative real numbers. The sum of this collection is defined to be the supremum of the set of all sums over finite subsets of gamma. This is

The sum of_[(alpha)(E)(gamma)]x_(alpha) = sup{(the sum of)_(alpha)(E)(S)x_(alpha): S is finite subset of gamma}
Prove that is the sum of[(alpha)(E)(gamma)x(alpha) < (infinity), the gamma is countable.

Hint:
Examining {(alpha)(E)(T):x_(alpha) >= 1/n} may be fruitful

Please see attachment for notation.

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This solution gives a step-by-step explanation on how to approach these kind of problems on countable sets. The solution is written in a half page pdf file.

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