16 1. Measure theory of a set E ⊂ Rd (we adopt the convention that m∗,(J)(E) = +∞ if E is unbounded, thus m∗,(J) now takes values in the extended non-negative reals [0, +∞], whose properties we will briefly review below). Observe from the finite additivity and subadditivity of elementary measure that we can also write the Jordan outer measure as m∗,(J)(E) := inf B1∪...∪Bk⊃E B1,...,Bk boxes |B1| + . . . + |Bk|, i.e., the Jordan outer measure is the infimal cost required to cover E by a finite union of boxes. (The natural number k is allowed to vary freely in the above infimum.) We now modify this by replacing the finite union of boxes by a countable union of boxes, leading to the Lebesgue outer measure9 m∗(E) of E: m∗(E) := inf ∞ n=1 Bn⊃E B1,B2,... boxes ∞ n=1 |Bn|, thus the Lebesgue outer measure is the infimal cost required to cover E by a countable union of boxes. Note that the countable sum ∑∞ n=1 |Bn| may be infinite, and so the Lebesgue outer measure m∗(E) could well equal +∞. Clearly, we always have m∗(E) ≤ m∗,(J)(E) (since we can always pad out a finite union of boxes into an infinite union by adding an infinite number of empty boxes). But m∗(E) can be a lot smaller: Example 1.2.1. Let E = {x1,x2,x3,...} ⊂ Rd be a countable set. We know that the Jordan outer measure of E can be quite large for instance, in one dimension, m∗,(J)(Q) is infinite, and m∗,(J)(Q ∩ [−R, R]) = m∗,(J)([−R, R]) = 2R since Q ∩ [−R, R] has [−R, R] as its closure (see Exercise 1.1.18). On the other hand, all countable sets E have Lebesgue outer measure zero. Indeed, one simply covers E by the degenerate boxes {x1}, {x2},... of sidelength and volume zero. Alternatively, if one does not like degenerate boxes, one can cover each xn by a cube Bn of sidelength ε/2n (say) for some arbitrary ε 0, leading to a total cost of ∑ ∞ n=1 (ε/2n)d, which converges to Cdεd for some absolute constant Cd. As ε can be arbitrarily small, we see that the Lebesgue outer measure must be zero. We will refer to this type of trick as the ε/2n trick it will be used many more times in this text. From this example we see, in particular, that a set may be unbounded while still having Lebesgue outer measure zero, in contrast to Jordan outer measure. 9Lebesgue outer measure is also denoted m∗(E) in some texts.

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