18 1. Measure theory as almost every set one actually encounters in analysis will be measurable (the main exceptions being some pathological sets that are constructed using the axiom of choice). In the next set of notes we will use Lebesgue mea- sure to set up the Lebesgue integral, which extends the Riemann integral in the same way that Lebesgue measure extends Jordan measure and the many pleasant properties of Lebesgue measure will be reflected in analogous pleasant properties of the Lebesgue integral (most notably the convergence theorems). We will treat all dimensions d = 1, 2,... equally here, but for the pur- poses of drawing pictures, we recommend to the reader that one sets d equal to 2. However, for this topic at least, no additional mathematical diﬃculties will be encountered in the higher-dimensional case (though of course there are significant visual diﬃculties once d exceeds 3). 1.2.1. Properties of Lebesgue outer measure. We begin by studying the Lebesgue outer measure m∗, which was defined earlier, and takes values in the extended non-negative real axis [0, +∞]. We first record three easy properties of Lebesgue outer measure, which we will use repeatedly in the sequel without further comment: Exercise 1.2.3 (The outer measure axioms). (i) (Empty set) m∗(∅) = 0. (ii) (Monotonicity) If E ⊂ F ⊂ Rd, then m∗(E) ≤ m∗(F ). (iii) (Countable subadditivity) If E1,E2,... ⊂ Rd is a countable se- quence of sets, then m∗( ∞ n=1 En) ≤ ∑ ∞ n=1 m∗(En). (Hint: Use the axiom of countable choice, Tonelli’s theorem for series, and the ε/2n trick used previously to show that countable sets have outer measure zero.) Note that countable subadditivity, when combined with the empty set axiom, gives as a corollary the finite subadditivity property m∗(E1 ∪ . . . ∪ Ek) ≤ m∗(E1) + . . . + m∗(Ek) for any k ≥ 0. These subadditivity properties will be useful in establishing upper bounds on Lebesgue outer measure. Establishing lower bounds will often be a bit trickier. (More generally, when dealing with a quantity that is defined using an infimum, it is usually easier to obtain upper bounds on that quantity than lower bounds, because the former requires one to bound just one element of the infimum, whereas the latter requires one to bound all elements.) Remark 1.2.4. Later on in this text, when we study abstract measure the- ory on a general set X, we will define the concept of an outer measure on

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