By Terence Tao
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Additional info for An Introduction To Measure Theory (January 2011 Draft)
In , such that f is equal to a constant ci on each of the intervals Ii . If f is piecewise constant, show that the expression n ci |Ii | i=1 is independent of the choice of partition used to demonstrate the piecewise constant nature of f . c. a f (x) dx, and refer to it as the piecewise constant integral of f on [a, b]. 21 (Basic properties of the piecewise constant integral). Let [a, b] be an interval, and let f, g : [a, b] → R be piecewise constant functions. c. c. c. c. c. b a g(x) dx. e. c.
Non-measurable sets. In the previous section we have set out a rich theory of Lebesgue measure, which enjoys many nice properties when applied to Lebesgue measurable sets. Thus far, we have not ruled out the possibility that every single set is Lebesgue measurable. There is good reason for this: a famous theorem of Solovay[So1970] asserts that, if one is willing to drop the axiom of choice, there exist models of set theory in which all subsets of Rd are measurable. So any demonstration of the existence of nonmeasurable sets must use the axiom of choice in some essential way.
However, this is not the most intuitive formulation of this concept to work with, and we will instead use a different (but logically equivalent) definition of Lebesgue measurability. 13) that Jordan measurable sets can be efficiently contained in elementary sets, with an error that has small Jordan outer measure. 2 (Lebesgue measurability). A set E ⊂ Rd is said to be Lebesgue measurable if, for every ε > 0, there exists an open set U ⊂ Rd containing E such that m∗ (U \E) ≤ ε. If E is Lebesgue measurable, we refer to m(E) := m∗ (E) as the Lebesgue measure of E (note that this quantity may be equal to +∞).
An Introduction To Measure Theory (January 2011 Draft) by Terence Tao