By Terence Tao
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Extra resources for An Introduction To Measure Theory (January 2011 Draft)
5. Show that if a set E ⊂ Rd is expressible as the countable union of almost disjoint boxes, then the Lebesgue outer measure of E is equal to the Jordan inner measure: m∗ (E) = m∗,(J) (E), where we extend the definition of Jordan inner measure to unbounded sets in the obvious manner. Not every set can be expressed as the countable union of almost disjoint boxes (consider for instance the irrationals R\Q, which contain no boxes other than the singleton sets). 11. Let E ⊂ Rd be an open set. Then E can be expressed as the countable union of almost disjoint boxes (and, in fact, as the countable union of almost disjoint closed cubes).
Iii) Every set of Lebesgue outer measure zero is measurable. ) (iv) The empty set ∅ is Lebesgue measurable. (v) If E ⊂ Rd is Lebesgue measurable, then so is its complement Rd \E. (vi) If E1 , E2 , E3 , . . ⊂ Rd are a sequence of Lebesgue measur∞ able sets, then the union n=1 En is Lebesgue measurable. (vii) If E1 , E2 , E3 , . . ⊂ Rd are a sequence of Lebesgue measur∞ able sets, then the intersection n=1 En is Lebesgue measurable. Proof. Claim (i) is obvious from definition, as are Claims (iii) and (iv).
Thus E is not Lebesgue measurable. Unfortunately, the above argument is terribly non-rigorous for a number of reasons, not the least of which is that it uses an uncountable number of coin flips, and the rigorous probabilistic theory that one would have to use to model such a system of random variables is too weak12 to be able to assign meaningful probabilities to such events as “E is Lebesgue measurable”. So we now turn to more rigorous arguments that establish the existence of non-measurable sets.