1.2. Lebesgue measure 17 As we shall see in Section 1.7, Lebesgue outer measure (also known as Lebesgue exterior measure) is a special case of a more general concept known as an outer measure. In analogy with the Jordan theory, we would also like to define a concept of “Lebesgue inner measure” to complement that of outer measure. Here, there is an asymmetry (which ultimately arises from the fact that elemen- tary measure is subadditive rather than superadditive): one does not gain any increase in power in the Jordan inner measure by replacing finite unions of boxes with countable ones. But one can get a sort of Lebesgue inner mea- sure by taking complements see Exercise 1.2.18. This leads to one possible definition for Lebesgue measurability, namely the Carath´ eodory criterion for Lebesgue measurability see Exercise 1.2.17. However, this is not the most intuitive formulation of this concept to work with, and we will instead use a different (but logically equivalent) definition of Lebesgue measurability. The starting point is the observation (see Exercise 1.1.13) that Jordan measur- able sets can be eﬃciently contained in elementary sets, with an error that has small Jordan outer measure. In a similar vein, we will define Lebesgue measurable sets to be sets that can be eﬃciently contained in open sets, with an error that has small Lebesgue outer measure: Definition 1.2.2 (Lebesgue measurability). A set E ⊂ Rd is said to be Lebesgue measurable if, for every ε 0, there exists an open set U ⊂ Rd containing E such that m∗(U\E) ≤ ε. If E is Lebesgue measurable, we refer to m(E) := m∗(E) as the Lebesgue measure of E (note that this quantity may be equal to +∞). We also write m(E) as md(E) when we wish to emphasise the dimension d. Remark 1.2.3. The intuition that measurable sets are almost open is also known as Littlewood’s first principle, this principle is a triviality with our current choice of definitions, though less so if one uses other, equivalent, def- initions of Lebesgue measurability. See Section 1.3.5 for a further discussion of Littlewood’s principles. As we shall see later, Lebesgue measure extends Jordan measure, in the sense that every Jordan measurable set is Lebesgue measurable, and the Lebesgue measure and Jordan measure of a Jordan measurable set are always equal. We will also see a few other equivalent descriptions of the concept of Lebesgue measurability. In the notes below we will establish the basic properties of Lebesgue mea- sure. Broadly speaking, this concept obeys all the intuitive properties one would ask of measure, so long as one restricts attention to countable opera- tions rather than uncountable ones, and as long as one restricts attention to Lebesgue measurable sets. The latter is not a serious restriction in practice,

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