Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 26J / c

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- Solution c

Solution

The rational numbers are countable, so

Q = ⋃_{q \in Q} {q}

is a countable union of closed sets and hence a Borel set; its Lebesgue measure is zero. Its complement is also Borel and has infinite measure:

m (Q) = 0, m (R ∖ Q) = \infty.

The intervals are nested in the relevant directions, and the strongest lower and upper restrictions both occur at

n = 2

:

⋂_{n \geq 2} [1/ n, n] = [1/2, 2] .

Thus this set is closed, hence Borel, and

m (n \geq 2 ⋂ [1/ n, n]) = \frac{3}{2} .

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