Exercise 0.0.1 (Sum of uncountable number of non-negative numbers)

Question

If $(x_\alpha)_{\alpha \in A}$ is a collection of numbers $x_\alpha \in [0,+\infty]$ such that $\sum_{\alpha\in A} x_\alpha < \infty$, show that $x_\alpha = 0$ for all but at most countably many $\alpha \in A$, even if $A$ itself is uncountable.

DeeThreeCay · Accepted Answer

Assume on the contrary that $x_\alpha > 0$ for uncountably many $x_\alpha > 0$. \
Define $S_n = \{\alpha: x_\alpha \geq \frac{1}{n}\}$, $|S_n| = \infty$ for some choice of $n \in \mathbb{N}$ (else, we can show that $\bigcup_{i=1}^{\infty} S_i$ is countable) \
$\sum_\alpha x_\alpha \geq \sum_{S_n} \frac{1}{n} = \infty$, contradicting the fact that the series is finite.

More explanations at: \url{https://math.stackexchange.com/questions/20661/the-sum-of-an-uncountable-number-of-positive-numbers}

Exercise 0.0.1 (Sum of uncountable number of non-negative numbers)

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Exercise 0.0.1 (Sum of uncountable number of non-negative numbers)

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