An Introduction to Functional Analysis (North-Holland Texts - download pdf or read online

By Mischa Cotlar

ISBN-10: 0720420660

ISBN-13: 9780720420661

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Extra resources for An Introduction to Functional Analysis (North-Holland Texts in Advanced Mathematics)

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3 (5), it is the σ-algebra of subsets of R generated by open intervals, and null sets, or the one generated by closed intervals and null sets. 1. (Lebesgue measurable set). The σ-algebra of subsets of R generated by open intervals and null sets will be denoted by M. Sets in M will be called Lebesgue measurable, or measurable for short. If I is a closed interval, then M(I) will denote the Lebesgue measurable subsets of I. For simplicity we will focus on subsets of I = [0, 1] though we could just as well use any other interval.

If f is a continuous function on [a, b] and F is any anti-derivative of f, then b a f (x) dx = F (b) − F (a). x Proof. Define the function G(x) = a f (t) dt. 1 the derivative of G(x) is f (x) which is also the derivative of F . 4). Then F (b) − F (a) = (G(b) + C) − (G(a) + C) = G(b) − G(a) b a f (x) dx − = a f (x) dx a b f (x) dx. 3. (1) Prove that if f : [a, b] → R is a regulated function and x F : [a, b] → R is defined by F (x) = a f (t) dt, then F is continuous. (2) Let S denote the set of all functions F : [a, b] → R which can x be expressed as F (x) = a f (t) dt for some step function f.

3) If A and B are measurable subsets of [0, 1], prove that µ(A) + µ(B) = µ(A ∪ B) + µ(A ∩ B). (4) Prove that if X ⊂ I is measurable, then for any ε > 0 there is an open set U containing X such that µ(U \ X) < ε. This is sometimes referred to as the first of Littlewood’s three principles. 5. The Lebesgue Density Theorem 35 (5) Suppose a < b and let M([a, b]) denote the Lebesgue measurable subsets of [a, b]. Define the function f : [0, 1] → [a, b] by f (x) = mx + a where m = b − a. Show that the correspondence A → f (A) is a bijection from M([0, 1]) to M([a, b]).

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An Introduction to Functional Analysis (North-Holland Texts in Advanced Mathematics) by Mischa Cotlar


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