Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 1 / 26J / b

Table of contents
- Solution b

Solution

Let

f_{n}

be Borel measurable and converge pointwise to the finite real value

f

. For every real

a

,

{f > a} = r \in Q r > a ⋃ N \geq 1 ⋃ n \geq N ⋂ {f_{n} > r} .

If

f (x) > a

, choose a rational

r

strictly between them; convergence puts

f_{n} (x) > r

eventually. Conversely eventual strict inequality

f_{n} (x) > r > a

implies

f (x) \geq r > a

. The right side belongs to

E

by its countable union/intersection operations. Since these half-lines generate the Borel sigma-algebra,

f

is a Borel measurable function.

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