chapter one

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,
               ...