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Guidance on problem regarding non-atomic probability space

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Let's do proof by contradiction. Suppose that for every B⊆A we have P(B) >= e. By the definition of atomic, P(A) > 0 implies a B⊂A s.t. 0 < P(B) < P(A). Pick such a set B. By our starting assumption P(B) >= e. Consider the set A/B. This set is also a subset of A and so by our starting assumption also has probability equal to or  higher than e.

We can repeat this process for B. Pick a set C such that C⊂B and 0 < P(C) < P(B). The set B/C also has probability equal to or higher than e by assumption, and is disjoint from A/B, so P(A/B U B/C) >= 2e.

This process can be repeated any number of times to create an arbitrarily large set of distinct events each with probability at least e. This means we can find a subset of A with arbitrarily large probability. But the total probability measure of a set can not be greater than 1, so this is a contradiction.

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