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

By Terence Tao

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

**Example text**

7, Lebesgue outer measure (also known as Lebesgue exterior measure) is a special case of a more general concept known as an outer measure. In analogy with the Jordan theory, we would also like to define a concept of “Lebesgue inner measure” to complement that of outer measure. Here, there is an asymmetry (which ultimately arises from the fact that elementary measure is subadditive rather than superadditive): one does not gain any increase in power in the Jordan inner measure by replacing finite unions of boxes with countable ones.

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. g. A = [0, 1]d ). (i) Let 2A := {E : E ⊂ A} be the power set of A. We say that two sets E, F ∈ 2A are equivalent if E∆F is a null set. Show that this is a equivalence relation. (ii) Let 2A / ∼ be the set of equivalence classes [E] := {F ∈ 2A : E ∼ F } of 2A with respect to the above equivalence relation.

The set n=1 Un , being a union of open sets, is itself open, and the claim follows. Now we establish Claim (ii). Every closed set E is the countable union of closed and bounded sets (by intersecting E with, say, the closed balls B(0, n) of radius n for n = 1, 2, 3, . ), so by (vi), it suffices to verify the claim when E is closed and bounded, hence compact by the Heine-Borel theorem. Note that the boundedness of E implies that m∗ (E) is finite. Let ε > 0. 12), we can find an open set U containing E such that m∗ (U ) ≤ m∗ (E) + ε.