 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2011/d/d4/a/solution

0 \leq f_{1} \leq f_{2} \leq \dots

converging pointwise to

f

, the limit is measurable and

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

Both sides may be infinite. Monotonicity and nonnegativity replace any need for an integrable dominating function.

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