Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 6 / a / iii

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- Solution iii

Solution

The integers form a countable set, and the events

{X = k}

are disjoint. By countable additivity and part ii,

P (X \in Z) = k \in Z \sum P (X = k) = 0.

Countability is essential: one cannot use the same sum argument over an arbitrary uncountable collection of zero-probability events.

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Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 6 / b / v / Solution

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