3. Prove that if f is a real function on a measurable space X such that
{x : f (x) ≥ r} is measurable for every rational r, then f is measurable.
[proof]:For each real number α, there exists an descending sequence {rn }
of rational numbers such that lim rn = α. Moreover,we have
n→∞
[
∞
(α, +∞) = [rn , +∞).
n=1
Hence,
...
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