 Solution

ID: past-exam-of-the-options-ii-for-mathematics-and-philosophy-course-of-the-university-of-oxford/2013/d4/e/ii/solution

No such sequence exists. By Fatou's lemma on

(0, 1)

\int_{0}^{1} lim inf_{n} ∣ g_{n} (x) ∣ d x \leq lim inf_{n} \int_{0}^{1} ∣ g_{n} (x) ∣ d x \leq lim inf_{n} ∥ g_{n} ∥_{1} = 0.

But the proposed nonnegative measurable integrand is positive at every point of

(0, 1)

, so its integral cannot be zero. Explicitly, the sets where it is at least

1/ k

cover

(0, 1)

; at least one has positive measure, forcing a positive integral. This contradicts the displayed bound.

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