28 1. Measure theory Finally, Claim (vii) follows from (v), (vi), and de Morgan’s laws ( α∈A Eα)c = α∈A Eα, c ( α∈A Eα)c = α∈A Eα,c (which work for infinite unions and intersections without any diﬃculty). Informally, the above lemma asserts (among other things) that if one starts with such basic subsets of Rd as open or closed sets and then takes at most countably many Boolean operations, one will always end up with a Lebesgue measurable set. This is already enough to ensure that the ma- jority of sets that one actually encounters in real analysis will be Lebesgue measurable. (Nevertheless, using the axiom of choice one can construct sets that are not Lebesgue measurable we will see an example of this later. As a consequence, we cannot generalise the countable closure properties here to uncountable closure properties.) Remark 1.2.14. The properties (iv), (v), (vi) of Lemma 1.2.13 assert that the collection of Lebesgue measurable subsets of Rd form a σ-algebra, which is a strengthening of the more classical concept of a Boolean algebra. We will study abstract σ-algebras in more detail in Section 1.4. Note how Lemma 1.2.13 is significantly stronger than the counterpart for Jordan measurability (Exercise 1.1.6), in particular, by allowing countably many Boolean operations instead of just finitely many. This is one of the main reasons why we use Lebesgue measure instead of Jordan measure. Exercise 1.2.7 (Criteria for measurability). Let E ⊂ Rd. Show that the following are equivalent: (i) E is Lebesgue measurable. (ii) (Outer approximation by open) For every ε 0, one can contain E in an open set U with m∗(U\E) ≤ ε. (iii) (Almost open) For every ε 0, one can find an open set U such that m∗(UΔE) ≤ ε. (In other words, E differs from an open set by a set of outer measure at most ε.) (iv) (Inner approximation by closed) For every ε 0, one can find a closed set F contained in E with m∗(E\F ) ≤ ε. (v) (Almost closed) For every ε 0, one can find a closed set F such that m∗(F ΔE) ≤ ε. (In other words, E differs from a closed set by a set of outer measure at most ε.) (vi) (Almost measurable) For every ε 0, one can find a Lebesgue measurable set Eε such that m∗(EεΔE) ≤ ε. (In other words, E differs from a measurable set by a set of outer measure at most ε.)

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