 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2010/d/d1/2/solution

For every real

a

{x \in E : - g (x) \geq a} = E ∖ {x \in E : g (x) > - a} .

Part (1), applied to measurable

g

, makes the set being complemented measurable. Its relative complement in measurable

E

is measurable too. The permitted half-line criterion therefore proves $- g$ is measurable.

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