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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. 18. 17. 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.

Qk ) = m(Q1 ∪ . . ∪ Qk ) = m(Q1 ) + . . + m(Qk ) ≥ m(Q1 ) + . . + m(Qk ) − ε = m(E) − ε. 2. Lebesgue measure 27 Applying by monotonicity of Lebesgue outer measure, we conclude that m∗ (E) ≥ m(E) − ε for every ε > 0. Since ε > 0 was arbitrary, the claim follows. The above lemma allows us to compute the Lebesgue outer measure of a finite union of boxes. From this and monotonicity we conclude that the Lebesgue outer measure of any set is bounded below by its Jordan inner measure. 2) for every E ⊂ Rd .

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. c. a g(x) dx. c. a 1E (x) dx = m(E). 6 (Darboux integral). Let [a, b] be an interval, and f : [a, b] → R be a bounded function. c. piecewise constant g(x) dx, a where g ranges over all piecewise constant functions that are pointwise bounded above by f .

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An introduction to measure theory by Tao T.

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Categories: Calculus