Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 204 / 2 / e

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- Solution e

Solution

Within the monotone coupling,

I_{p}

increases with

p

and

⋃_{p^{'} < p} I_{p^{'}} = {M < p} .

Taking a countable cofinal sequence

p^{'} ↑ p

and using continuity from below of a measure gives

lim_{p^{'} ↑ p} θ (p^{'}) = P (M < p) .

Since

{M < p} \subseteq I_{p}

, subtraction from

θ (p) = P (I_{p})

proves

θ (p) - p^{'} ↑ p lim θ (p^{'}) = P (I_{p} \cap {M = p}) .

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