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Extra resources for An Introduction To Measure Theory (January 2011 Draft)

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Iii) For every box B, one has |B| = m∗ (B ∩ E) + m∗ (B\E). 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. e. that if A, A are two elementary sets containing E, that 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. ∞ Define a Gδ set to be a countable intersection n=1 Un of open ∞ sets, and an Fσ set to be a countable union n=1 Fn of closed sets.

It is convenient to introduce the following notion: two boxes are almost disjoint if their 28 1. Measure theory interiors are disjoint, thus for instance [0, 1] and [1, 2] are almost disjoint. 3) m(B1 ∪ . . ∪ Bk ) = |B1 | + . . + |Bk | holds for almost disjoint boxes B1 , . . , Bk , and not just for disjoint boxes. 9 (Outer measure of countable unions of almost disjoint ∞ boxes). Let E = n=1 Bn be a countable union of almost disjoint boxes B1 , B2 , . .. Then ∞ m∗ (E) = |Bn |. n=1 Thus, for instance, Rd itself has an infinite outer measure.

Show that there exists disjoint bounded subsets E, F of the real line such that m∗ (E ∪ F ) = m∗ (E) + m∗ (F ). 27 (Projections of measurable sets need not be measurable). Let π : R2 → R be the coordinate projection π(x, y) := x. Show that there exists a measurable subset E of R2 such that π(E) is not measurable. 46 1. 19. The above discussion shows that, in the presence of the axiom of choice, one cannot hope to extend Lebesgue measure to arbitrary subsets of R while retaining both the countable additivity and the translation invariance properties.

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An Introduction To Measure Theory (January 2011 Draft) by Terence Tao

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