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Lecture Notes on Measure Theory and Integration 1 — Introduction

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Lecture Notes on Measure Theory and Integration 1 — Introduction Lecture Notes on Measure Theory and Integration 1 — Introduction Joel Feldman University of British Columbia September 4, 2019 1 1 Introduction 1.1 What are “Measure Theory and the Lebesgue Integral”? b In introductory Calculus courses a f(x) dx is invariably defined to be the Riemann integral. The Lebesgue integral is an alternative definition. For simplicity, assume R that a < b and f(x) ≥ 0. b • Under the Riemann integral definition of a f(x) dx, ◦ for each n ∈ N, we subdivide the domain of integration into n (for example, R equal) subintervals and correspondingly subdivide R = (x, y) a ≤ x ≤ b, 0 ≤ y ≤ f(x) into n vertical strips (x, y) x i−1 ≤ x ≤ xi, 0 ≤ y ≤ f(x) , 1 ≤ i ≤ n. y y = f(x) x x0 = a x1 x2 x3 x4 x5 x6 x7 x8 = b ∗ th ◦ Then for each 1 ≤ i ≤ n, we pick some xi in the i subinterval and approximate the area of the ith strip by the area of the rectangle which has the same width ∗ as the strip and which has height f(xi ). n ∗ ◦ In this way we approximate the area of R by i=1 f(xi )(xi − xi−1). ◦ By definition, the Riemann integral b f(x) dx is the limit of this expression as a P n →∞. R Introduction 2 September 4, 2019 b • Under the Lebesgue integral definition of a f(x) dx, ◦ for each n ∈ N, we subdivide the y-axis, rather than the x-axis, into subintervals 1 R of length n and correspondingly subdivide R = (x, y) a ≤ x ≤ b, 0 ≤ y ≤ f(x) i i+1 into pieces Ri = (x, y) ∈ R n ≤ f(x) ≤ n y y = f(x) (i+1)/n i/n Xi Xi Xi Xi x ◦ Denote by − 1 i (i+1) Xi = f [ /n, /n) ∩ [a, b] i (i+1) the set of x’s in [a, b] were f takes values in the interval [ /n, /n) . On Xi we approximate f by i . So we approximate the area of R by the height i , n i n times its width, which is the “size” of Xi, given by the sum of the lengths of the various components of Xi. We use m(Xi) to denote that size, which we call the “measure” of Xi. i ◦ In this way we approximate the area of R by i n m(Xi). b ◦ By definition, the Lebesgue integral a f(x) dxPis the limit of this expression as n →∞. R ◦ The main step in filling out this outline of the definition of the Lebesgue integral is to make precise what we mean by “the size of a set”. That’s “measure theory”. Introduction 3 September 4, 2019 1.2 Why Bother with Measure Theory? • We will eventually1 see the following theorem, which gives us some info regarding the relationship between the Riemann and Lebesgue integrals. Theorem. Let a < b be real numbers and let f :[a, b] → R be bounded. (a) If f is Riemann integrable, then f is Lebesgue measurable and a R f(x) dx = f(x) dm(x) Zb Z[a,b] (b) f is Riemann integrable if and only if x ∈ [a, b] f is not continuous at x has Lebesgue measure zero. In part (a), the integral on the left is the Riemann integral and the integral on the right is the Lebesgue integral. We have not yet defined “Lebesgue measure zero”. Any set which contains at most countable many points has Lebesgue measure zero. • The measure theory approach to integration is much more technically demanding than the Riemann integral approach, and the Riemann integral works perfectly well with even slightly smooth integrands. So if you work only with, for exam- ple, piecewise continuous functions — and that is the case for much of “classical” applied mathematics and “classical” physics (for example classical mechanics, elec- tricity and magnetism, fluid dynamics) — there is no point is learning measure theory. • But if you have to deal with very irregular functions — and that is the case in quantum mechanics and option pricing using stochastic differential equations — the Lebesgue integral often works well even when the Riemann integral crashes and burns. • For example view 1 2 C 2 LR([0, 1]) = f : [0, 1] → f is Riemann integrable, |f(x)| dx < ∞ 0 n Z 1 o 2 C 2 LL([0, 1]) = f : [0, 1] → f is Lebesgue integrable, |f(x)| dx < ∞ 0 n Z o as metric spaces with the metric 2 1 1/2 d(f,g)= |f(x) − g(x)|2 dx Z0 1It is Theorem 4.15. 2To be picky, d(f,g) isn’t a metric, because there are nonzero functions f with d(f, 0) = 0. This is a technicality which is easy to deal with. Introduction 4 September 4, 2019 2 3 2 2 Then LR([0, 1]) is not complete , while LL([0, 1]) is complete. L spaces are per- vasive in quantum mechanics. • The measure theory approach to integration is easy to generalize. For example one can rigorously implement the “Dirac delta function” 0 if x =06 δ(x)= with δ(x) dx =1 +∞ if x =0 ( ) Z by making a suitable definition of the “size of a set”. In fact, our framework will be flexible to define an integral over any set, not just subsets of Rn. 3See the notes “Incompleteness of L2 with the Riemann Integral” for an example of a Cauchy sequence that does not converge. Introduction 5 September 4, 2019.
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