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Real Analysis Real Analysis 0 Introduction Dr Nikolai Chernov1 R b In Calculus, a definite integral a f(x) dx is computed by dividing the interval I = [a; b] into small subintervals I1;:::;In and approximating Z b Xn f(x) dx ≈ f(xi) jIij a i=1 j j Here Ii denotes the length of Ii and xi is any pointR in Ii. The limit, as the lengths b of each Ii approaches zero, gives the exact value of a f(x) dx. Examples:R In mechanics, if f(x) is the density (mass per unit length) of a rod [a; b] then b f(x) dx is the total mass of the rod. In probability theory, if f(x) is a R b the density of a 1D probability distribution then a f(x) dx is the total probability on the interval [a; b]. RR Similarly, a definite double integral R f(x; y) dx dy over a rectangle R = [a1; b1] × [a2; b2] is computed by partitioning R into small subrectangles R1;:::;Rn , choosing any (xi; yi) 2 Ri for i = 1; :::; n , and approximating ZZ Xn f(x; y) dx dy ≈ f(xi; yi) Area(Ri); R i=1 Here (xi; yi) is any pointRR in Ri. The limit, as the area of each Ri approaches zero, gives the exact value of R f(x; y) dx dy Examples: In mechanics,RR if f(x; y) is the density (mass per unit area) of a rect- angular plate R, then R f(x; y) dx dy is the total mass of the plate.RR In probability, if f(x; y) is the density of a 2D probability distribution, then R f(x; y) dx dy is the total probability over the rectangle R. 1Formatting and illustrations by Michael Pogwizd. 1 In the studies of probabilities, we often need to replace a 1D interval or 2D rectangle with other objects that can be more complicated. So it becomes necessary to generalize the notion of definite integral. What is common between the above two examples? A function f has a domain D (an interval or a rectangle) that can be divided into smaller pieces Di, and each piece is measured by a positive number (its length or area), the smaller the piece the smaller its measure. The measure of the whole domain D is the sum of measures of its pieces: Xn µ(D) = µ(Di) i=1 where µ denotes that measure (the length or area). The integral can then be approximated by Xn f(zi) µ(Di) i=1 where zi is a point in Di. The purpose of Real Analysis is to develop a machinery (a theoretical appara- tus) for integrating functions on arbitrary domains. We will generalize the \length of a line interval" and the \area of a rectangle" to an abstract notion of a \mea- sure of an arbitrary set". And we will learn how to integrate functions on arbitrary domains. The following chart summarizes the basic goals and terms of Real Analysis: 1D 2D General Interval I Rectangle R =) Space X Length Area =) Measure µ RIntegral RRDouble Integral ) LebesgueR Integral b = a f dx R f(x; y) dx dy X f dµ The theory of (abstract) measures and Lebesgue integration provides a solid foundation for modern Probability Theory. A good knowledge of Real Analysis is an absolute must for anyone who plans to do research in Dynamical Systems or Mathematical Physics. 2 1 Lebesgue measure in R and R2: basic constructions Definition 1.1. An interval I ⊂ R is a set of the form [a; b] or [a; b) or (a; b] or (a; b), where a ≤ b are real numbers. Its length is jIj = b − a • Including or excluding the endpoints of the interval does not affect its length. • Notation: A ] B always means disjoint union of two sets A and B i.e., such that \ ; ]1 A B = . Furthermore, n=1An always means the union of pairwise disjoint sets An, i.e., such that An \ Am = ; for all m =6 n. P ]N j j N j j Lemma 1.2. If I = n=1In, then I = n=1 In Proof. If N = 2 and c is the common endpoint of I1 and I2, then jI1j + jI2j = (b − c) + (c − a) = b − a = jIj. For N ≥ 2 use induction. Definition 1.3. A (linear) elementary set isP a finite union of disjoint intervals. ]N j j N j j i.e. J = n=1In The total length of J is J = n=1 In • In are not necessarily adjacent intervals, there may be gaps in between. 2 Definition 1.4. A rectangle R ⊂ R is a set of the form R = I1 × I2, where I1;I2 are intervals. The area of a rectangle is Area(R) = jI1j × jI2j • We only consider rectangles with horizontal and vertical sides. • Including or excluding the sides of the rectangles does not affect their areas. P ]N N Lemma 1.5. If R = n=1Rn, then Area(R) = n=1 Area(Rn). Proof. For N = 2 the proof is a trivial calculation. For N ≥ 2 we first need to extend the sides of Rn's so that they run completely across R (see Figure 1). This causes partitioning of some of the Rn's into smaller rectangles, which can be done one by one and using the already proved version of the lemma for N = 2. In the end we get a partition of R by k1 ≥ 1 vertical lines and k2 ≥ 1 horizontal lines (like a grid). Now the proof is a simple calculation using Lemma 1.2. Figure 1: A partition of a rectangle into smaller rectangles. Inner sides are extended so that they run completely across R (top to bottom and left to right). 3 Definition 1.6. A (planar) elementary set is a finite union of disjointP rectan- ]n n gles, i.e. B = i=1Ri. The area of an elementary set is Area(B) = i=1 Area(Ri) Theorem 1.7. (a) The length of a linear elementary set does not depend on how it is partitioned into disjoint intervals. (b) The area of a planar elementary set does not depend on how it is partitioned into disjoint rectangles. [n [m 0 Proof. We prove part (b). Let B = i=1Ri and B = j=1Rj be two partitions of an elementary set B into rectangles. Note that for each pair i; j the intersection \ 0 Rij = Ri Rj is a rectangle (which maybe empty). Now by Lemma 1.5: Xn Xn Xm Xm Xn Xm 1:5 1:5 0 Area(Ri) = Area(Rij) = Area(Rij) = Area(Rj): i=1 i=1 j=1 j=1 i=1 j=1 The easier part (a) is left as an exercise. Corollary 1.8. ]n (a) If J = i=1Ji is a finite union of disjoint linear elementary sets, then Pn jJj = jJij i=1 ]n (b) If B = i=1Bi is a finite union of disjoint planar elementary sets, then Pn Area(B) = Area(Bi) i=1 Theorem 1.9. Finite unions, intersections and differences of elementary sets are elementary sets. Proof. A direct inspection. We omit details. • A countable union of elementary sets is not necessarily an elementary set. Exercise 1. Show that the open disk x2 + y2 < 1 is a countable union of planar elementary sets. Show that the closed disk x2 + y2 ≤ 1 is a countable intersection of planar elementary sets. 4 Definition 1.10. A ring is a nonempty collection of subsets of a set X closed under finite unions, intersections and differences. Definition 1.11. An algebra is a ring containing X itself. Example 1. Linear elementary sets J ⊂ R make a ring. Planar elementary sets B ⊂ R2 make a ring. Example 2. Linear elementary sets J ⊂ R do not make an algebra. If we consider all finite unions of finite intervals and infinite intervals, then we will get an algebra. (An infinite interval is a set [a; 1) or (a; 1) or (−∞; a] or (−∞; a), where a 2 R1. The real line R1 itself is also an infinite interval.) Theorem 1.12. (a) If a linear elementary set J is covered by intervals I1;:::;In Pn ⊂ [n j j ≤ j j (i.e. J i=1Ii), then J Ii i=1 (b) If a planar elementary set B is covered by rectangles R1;:::;Rn Pn ⊂ [n ≤ (i.e. B i=1Ri), then Area(B) Area(Ri) i=1 \ \ n [i−1 Proof. We prove (b). By Theorem 1.9, B1 = B R1 and Bi = B (Ri j=1Rj) are elementary sets. Obviously, Bi ⊂ Ri, hence Area(Bi) ≤ Area(Ri). Since Bi are ]n disjoint and B = i=1Bi, we have Xn Xn 1:8(b) Area(B) = Area(Bi) ≤ Area(Ri): i=1 i=1 The easier part (a) is left as an exercise. Next we generalize the concepts of length in R1 and area in R2. This can be done in parallel, as above, but for the sake of brevity we only do it for the 2D case (i.e., for the area in R2). The conversion of all our definitions and theorems to the simpler 1D case is left as an exercise. For convenience, we want to avoid infinite areas at early stages of our construc- tions. In this section we just fix a large finite rectangle X ⊂ R2 and only consider subsets A ⊂ X. Note that elementary sets B ⊂ X make an algebra (because X is an elementary set). 5 Definition 1.13.
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