Measurable functions

Abstract

The length of ℝ is unbounded above, i.e. ‘infinite’. To deal with this we defined Lebesgue measure for sets of infinite as well as finite measure. In order to handle functions between such sets comprehensively, it is convenient to allow functions which take infinite values: we take their range to be (part of) the ‘extended real line’ ̄ ℝ =[− ∞,∞], obtained by adding the ‘points at infinity’ −∞ and +∞ to ℝ. Arithmetic in this set needs a little care as already observed in Section 2.2: we assume that a+ ∞ = ∞ for all real a, a × ∞ = ∞ for a > 0, a × ∞ = −∞ for a < 0, ∞ × ∞ = ∞ and 0 × ∞ = 0, with similar definitions for −∞. These are all ‘obvious’ intuitively (except possibly 0 × ∞), and (as for measures) we avoid ever forming ‘sums’ of the form ∞ + (−∞). With these assumptions ‘arithmetic works as before’.