 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2010/d/d1/1/solution

For every real

b

f^{- 1} ((b, \infty)) = n = 1 ⋃ \infty f^{- 1} ([b + 1/ n, \infty)) .

Each set on the right is measurable by hypothesis. Closure of a sigma-algebra under countable unions proves the assertion.

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