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Intuitively, we want to build a set which has oscillatory structure even at arbitrarily fine scales. We will non-rigorously do this as follows. We will work inside the unit interval [0, 1]. For each x ∈ [0, 1], we imagine that we flip a coin to give either heads or tails (with an independent coin flip for each x), and let E ⊂ [0, 1] be the set of all the x ∈ [0, 1] for which the coin flip came up heads. We suppose for contradiction that E is Lebesgue measurable. g. 4]) in an extremely nonrigorous fashion, we thus expect m(E) to equal 1/2.

Ii) (Countable additivity) If E1 , E2 , . . ⊂ Rd is a countable se∞ quence of disjoint Lebesgue measurable sets, then m( n=1 En ) = ∞ n=1 m(En ). (iii) (Translation invariance) If E is Lebesgue measurable and x ∈ Rd , then m(E + x) = m(E). (iv) (Normalisation) m([0, 1]d ) = 1. Hint: First show that m must match elementary measure on elementary sets, then show that m is bounded by outer measure. 24 (Lebesgue measure as the completion of elementary measure). The purpose of the following exercise is to indicate how Lebesgue measure can be viewed as a metric completion of elementary measure in some sense.

23 (Uniqueness of Lebesgue measure). Show that Lebesgue measure E → m(E) is the only map from Lebesgue measurable sets to [0, +∞] that obeys the following axioms: (i) (Empty set) m(∅) = 0. (ii) (Countable additivity) If E1 , E2 , . . ⊂ Rd is a countable se∞ quence of disjoint Lebesgue measurable sets, then m( n=1 En ) = ∞ n=1 m(En ). (iii) (Translation invariance) If E is Lebesgue measurable and x ∈ Rd , then m(E + x) = m(E). (iv) (Normalisation) m([0, 1]d ) = 1. Hint: First show that m must match elementary measure on elementary sets, then show that m is bounded by outer measure.

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An introduction to measure theory by Tao T.


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