 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2007/d/d1/iii/solution

g_{n} = x^{1/2} 1_{[1/ n, 1]}

are measurable, nonnegative, and increase almost everywhere to

g

. As before their integrals agree with the corresponding Riemann integrals, giving

\int g_{n} = \int_{1/ n}^{1} x^{1/2} d x = \frac{2}{3} (1 - n^{- 3/2}) .

Monotone convergence therefore gives the finite value

\int g = 2/3

 New to topics? Read the docs here!