1.2. Lebesgue measure 33 (iii) (Almost open bounded) E differs from a bounded open set by a set of arbitrarily small Lebesgue outer measure. (In other words, for every ε 0 there exists a bounded open set U such that m∗(EΔU) ε.) (iv) (Inner approximation by compact) For every ε 0, one can find a compact set F contained in E with m∗(E\F ) ε. (v) (Almost compact) E differs from a compact set by a set of arbi- trarily small Lebesgue outer measure. (vi) (Almost bounded measurable) E differs from a bounded Lebesgue measurable set by a set of arbitrarily small Lebesgue outer measure. (vii) (Almost finite measure) E differs from a Lebesgue measurable set with finite measure by a set of arbitrarily small Lebesgue outer measure. (viii) (Almost elementary) E differs from an elementary set by a set of arbitrarily small Lebesgue outer measure. (ix) (Almost dyadically elementary) For every ε 0, there exists an integer n and a finite union F of closed dyadic cubes of sidelength 2−n such that m∗(EΔF ) ε. One can interpret the equivalence of (i) and (ix) in the above exercise as asserting that Lebesgue measurable sets are those which look (locally) “pixelated” at sufficiently fine scales. This will be formalised in later sections with the Lebesgue differentiation theorem (Exercise 1.6.24). Exercise 1.2.17 (Carath´ eodory criterion, one direction). Let E Rd. Show that the following are equivalent: (i) E is Lebesgue measurable. (ii) For every elementary set A, one has m(A) = m∗(A∩E)+m∗(A\E). (iii) For every box B, one has |B| = m∗(B E) + m∗(B\E). Exercise 1.2.18 (Inner measure). Let E Rd be a bounded set. Define the Lebesgue inner measure m∗(E) of E by the formula m∗(E) := m(A) m∗(A\E) for any elementary set A containing E. (i) Show that this definition is well defined, i.e., that if A, A are two elementary sets containing E, then m(A) m∗(A\E) is equal to m(A ) m∗(A \E). (ii) Show that m∗(E) m∗(E), and that equality holds if and only if E is Lebesgue measurable.
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