Measure, Integral and Probability pp 15-53 | Cite as

# Measure

## Abstract

The idea of a ‘negligible’ set relates to one of the limitations of the Riemann integral, as we saw in the previous chapter. Since the function *f* = 1_{ℚ} takes a non-zero value only on ℚ, and equals 1 there, the ‘area under its graph’ (if such makes sense) must be very closely linked to the ‘length’ of the set ℚ. This is why it turns out that we cannot integrate *f* in the Riemann sense: the sets ℚ and ℝ\ℚ are so different from intervals that it is not clear how we should measure their ‘lengths’ and it is clear that the ‘integral’ of *f* over [0, 1] *should* equal the ‘length’ of the set of rationals in [0, 1]. So how should we define this concept for more general sets?

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