Measure, Integral, Derivative: A Course on Lebesgue's Theory (Universitext)

Measure, Integral, Derivative: A Course on Lebesgue's Theory (Universitext)

Language: English

Pages: 156

ISBN: 1461471958

Format: PDF / Kindle (mobi) / ePub

This classroom-tested text is intended for a one-semester course in Lebesgue’s theory.  With over 180 exercises, the text takes an elementary approach, making it easily accessible to both upper-undergraduate- and lower-graduate-level students.  The three main topics presented are measure, integration, and differentiation, and the only prerequisite is a course in elementary real analysis.

In order to keep the book self-contained, an introductory chapter is included with the intent to fill the gap between what the student may have learned before and what is required to fully understand the consequent text. Proofs of difficult results, such as the differentiability property of functions of bounded variations, are dissected into small steps in order to be accessible to students. With the exception of a few simple statements, all results are proven in the text.  The presentation is elementary, where ?-algebras are not used in the text on measure theory and Dini’s derivatives are not used in the chapter on differentiation. However, all the main results of Lebesgue’s theory are found in the book.

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≤ m(G). Lemma 2.6. Let G be an open subset of a bounded closed set F . Then m(G) ≤ m(F ). Proof. Let a = inf F and b = sup F . The sets G and [a, b] \ F are open and disjoint. It is clear that G ∪ ([a, b] \ F ) ⊆ (a, b). Therefore, by Theorems 2.3 and 2.2, m(G) + m([a, b] \ F ) ≤ b − a, that is, by the definition of m(F ), m(G) + (b − a) − m(F ) ≤ b − a, and the result follows. In summary, we have the following theorem. Theorem 2.5. Let U and V be bounded sets of real numbers such that each of

+ ≤ |f | and 0 ≤ f − ≤ |f |, the functions f + , f − are integrable over E, by the monotonicity property of integration for nonnegative functions. Hence, f is integrable over E. Finally, f = E E f+ − E f− ≤ f+ + E f− = E E |f |, by the triangle inequality for real numbers and the linearity of integration for nonnegative functions. Now we extend the linearity and monotonicity properties of integration to arbitrary integrable functions. 86 3 Lebesgue Integration Theorem 3.21. Let f be

g(γ). There are two possible cases: 1. γ < z ≤ d. Then g(z) > g(γ) ≥ g(x), which contradicts our choice of γ, because z > γ = sup{y ∈ [x, d] : g(y) ≥ g(x)}. 2. z > d. Then g(z) > g(γ) ≥ g(x) > g(d). It follows that d is a shadow point which is false because d ∈ / E. These contradictions show that γ = d and hence g(x) ≤ g(d) for all x ∈ (c, d). Lemma 4.15. Let f be an increasing absolutely continuous function on [a, b]. If Z is a subset of [a, b] of measure zero, then f (Z) is also a set of

closed set. Then points a = inf F and b = sup F belong to the set F and the set [a,b] F = [a, b] \ F is open. Proof. Inasmuch as a is the infimum of F , every open interval containing a must intersect the set F . Hence a cannot belong to the open set F . It follows that a ∈ F . Similar argument shows that b ∈ F . Clearly, F ⊆ [a, b]. Since a, b ∈ F , we have [a, b] \ F = (a, b) \ F = (a, b) ∩ F, which is an open set because the set F is closed. 12 1 Preliminaries Theorem 1.11. The only subsets

}p∈P is summable. Then the double family {apq }(p,q)∈P ×Q is summable. Accordingly, the equalities in (1.4) hold. If P = Q = N, we have a convergent double sequence (apq ) which terms form an infinite matrix. This matrix and its row and column sums are shown in the diagram below. a11 a21 .. . ap1 .. . p ap1 a12 a22 .. . ap2 .. . p ap2 ··· ··· ··· ··· a1q a2q .. . apq .. . p apq ··· ··· ··· ··· a1q q a2q .. . q apq .. . p,q apq q Notes The section on sets and functions (Sect. 1.1) serves two

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