6 1. Measure theory where 1 N Z := { n N : n ∈ Z} and #A denotes the cardinality of a finite set A. Taking Cartesian products, we see that |B| = lim N→∞ 1 N d #(B ∩ 1 N Zd) for any box B, and in particular, that |B1| + . . . + |Bk| = lim N→∞ 1 N d #(E ∩ 1 N Zd). Denoting the right-hand side as m(E), we obtain the claim (ii). Exercise 1.1.2. Give an alternate proof of Lemma 1.1.2(ii) by showing that any two partitions of E into boxes admit a mutual refinement into boxes that arise from taking Cartesian products of elements from finite collections of disjoint intervals. Remark 1.1.3. One might be tempted now to define the measure m(E) of an arbitrary set E ⊂ Rd by the formula (1.1) m(E) := lim N→∞ 1 N d #(E ∩ 1 N Zd), since this worked well for elementary sets. However, this definition is not particularly satisfactory for a number of reasons. First, one can concoct examples in which the limit does not exist (Exercise!). Even when the limit does exist, this concept does not obey reasonable properties such as translation invariance. For instance, if d = 1 and E := Q ∩ [0, 1] := {x ∈ Q : 0 ≤ x ≤ 1}, then this definition give E a measure of 1, but would give the translate E + √ 2 := {x + √would 2 : x ∈ Q 0 ≤ x ≤ 1} a measure of zero. Nevertheless, the formula (1.1) will be valid for all Jordan measurable sets (see Exercise 1.1.13). It also makes precise an important intuition, namely that the continuous concept of measure can be viewed5 as a limit of the discrete concept of (normalised) cardinality. From the definitions, it is clear that m(E) is a non-negative real number for every elementary set E, and that m(E ∪ F ) = m(E) + m(F ) whenever E and F are disjoint elementary sets. We refer to the latter property as finite additivity by induction it also implies that m(E1 ∪ . . . ∪ Ek) = m(E1) + . . . + m(Ek) 5Another way to obtain continuous measure as the limit of discrete measure is via Monte Carlo integration, although in order to rigorously introduce the probability theory needed to set up Monte Carlo integration properly, one already needs to develop a large part of measure theory, so this perspective, while intuitive, is not suitable for foundational purposes.

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