 Solution

ID: past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2007/d/1/i/solution

f_{n}

are nonnegative and

f_{n} ↑ f

f

is measurable and

\int f d μ = n \to \infty lim \int f_{n} d μ,

allowing either side to be infinite. The same conclusion holds under almost-everywhere monotone convergence after changing values on a null set. No assumption that the integrals stay bounded is required.

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