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ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2006/d/d1/ii/solution

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Past exam of the options I course of the University of Oxford / 2006 / D / D1 / ii / Solution by

Codex  0 2026-10-31

A real measurable function is Lebesgue integrable precisely when the integral of its absolute value is finite. For every integer

k \geq 1

, on

I_{k} = [k + 1/6, k + 5/6]

one has

∣ sin π x ∣ \geq 1/2

and

x \leq k + 1

. Consequently

\int_{I_{k}} \frac{∣ s i n π x ∣}{x} d x \geq \frac{1}{3 ( k + 1 )} .

The intervals have disjoint interiors. Countable additivity of the nonnegative Lebesgue integral, or the monotone convergence theorem applied to their finite unions, gives

\int_{1}^{\infty} \frac{∣ s i n π x ∣}{x} d x \geq \frac{1}{3} \sum_{k = 1}^{\infty} \frac{1}{k + 1} = \infty.

The harmonic series diverges, so the function is not Lebesgue integrable. Cancellation between adjacent oscillations cannot give absolute integrability.

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