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Generalized Riemann Integrals: GR-Integrable Functions and Their Properties - Prof. David , Study notes of Mathematics

Notes on the concept of generalized riemann integrals, which extends the definition of riemann integrals to include functions that are not riemann-integrable but intuitively integrable. The definition of gr-integrable functions, their relationship with riemann integrals, and various properties such as continuity and boundedness. It also includes exercises to test understanding.

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Pre 2010

Uploaded on 03/16/2009

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Download Generalized Riemann Integrals: GR-Integrable Functions and Their Properties - Prof. David and more Study notes Mathematics in PDF only on Docsity! Notes by David Groisser, Copyright c©1995, revised version 2002 MAA 4212—Improper Integrals The Riemann integral, while perfectly well-defined, is too restrictive for many pur- poses; there are functions which we intuitively feel “ought” to be integrable, but which are not Riemann integrable according to the definition. For example, the expression∫ 1 0 1√ t dt makes no sense as a Riemann integral, since the integrand is not defined at t = 0. Even if we fix that problem, by defining a function that’s t−1/2 for t > 0 and (say) 0 for t = 0, this new function is still not Riemann-integrable over [0, 1] because it isn’t bounded. However, if formally make the change of variables t = u2 (“formally” means shoot first, question validity later), the integral above gets transformed into∫ 1 0 2 du, which is as nice an integral as they come. Furthermore, if we go back to our original integral and think of it as representing area under a curve, there is a useful sense in which this area is finite: take the area below the curve between t =  and t = 1, and let → 0. Either way of looking at the original integral, the answer we formally calculate is 2. These considerations suggest that we ought to enlarge the class of functions we’re willing to call “integrable”, and modify our definition of “integral”. The types of integrals we’ll deal with in this handout are often called “improper integrals”, but we’ll simply call them “integrals” here. Terminology. In this handout, the words “integral” and “integrable” will not be synonymous with “Riemann integral” and “Riemann-integrable”. (In Rosenlicht, they are synonymous, but we will need to be clearer here on what notion of integration we’re talking about.) We will use notation “Riemann ∫ b a f(t)dt” (with the word “Riemann” in front of the integral sign) to denote the Riemann integral. Whenever we write hypotheses such as “Let f : [a, b] → R”, we understand this as short-hand for “Let a, b ∈ R with a < b and let f : [a, b] → R;” analogous interpretations apply if [a, b] is replaced by (a, b], [a, b), or (a, b). Also, we write “limx↑a” and “limx↓a” in place of limx→a−, limx→a+ respectively. §1 Integrals over bounded intervals. Definition 1. We will say that a real-valued function f is GR-integrable (for “generalized Riemann integrable”) on the interval [a, b] if either (i) f is defined on (a, b] and is Riemann-integrable over [y, b] for all y ∈ (a, b], and limy↓a Riemann ∫ b y f(t)dt exists; or 1 (ii) f is defined on [a, b) and is Riemann-integrable over [a, y] for all y ∈ [a, b), and limy↑b Riemann ∫ y a f(t)dt exists. (This is a temporary definition that will be generalized and finalized in Definition 3.) Note that if f satisfies both conditions (i) and (ii), then it is Riemann-integrable over [a, b]. In particular, every Riemann-integrable function is GR-integrable. Note. The terms “generalized Riemann integral” and “GR-integrable” are specific to these notes; they are not standard terminology. Exercise. 1. Suppose f : [a, b] → R is Riemann-integrable on [a, b]. Prove that for all c ∈ [a, b], the functions g, h defined by g(x) = Riemann ∫ x c f(t)dt, h(x) = Riemann ∫ c x f(t) dt are continuous. In particular, Exercise 1 implies that if f is Riemann-integrable on [a, b], then lim y↓a [Riemann ∫ b y f(t)dt] = Riemann ∫ b a f(t)dt (1) and lim y↑b [Riemann ∫ y a f(t)dt] = Riemann ∫ b a f(t)dt. (2) This suggests using equations (1) and (2) to define the integral in certain non-Riemann- integrable cases. Definition 2. Let f : [a, b]→ R. If f satisfies condition (i) in Definition 1, we define the generalized Riemann integral∫ b a f(t)dt = lim y↓a [Riemann ∫ b y f(t)dt]. (3) Similarly if f satisfies condition (ii) in Definition 1, we define∫ b a f(t)dt = lim y↑b [Riemann ∫ y a f(t)dt]. (4) In both cases we will say that f is GR-integrable on [a, b]. Note. In place of saying “f is GR-integrable”, we often say “the integral exists” or “the integral converges”. Equations (1–2) show that there is no ambiguity in Definition 2; if f satisfies both (i) and (ii) in Definition 1, then the limits in equations (3) and (4) are equal. Moreover, if f is Riemann-integrable on [c, d] then ∫ d c f(t)dt = Riemann ∫ d c f(t)dt. Hence if f is GR-integrable on [a, b] we may write equations (3) and (4) more simply as 2 ∫ b a f(t)dt = ∫ c a f(t)dt+ ∫ b c f(t)dt. (6) (End of exercise 7.) Now suppose we have a function f defined on [a, b], continuous except for an interior singularity at a single point c ∈ (a, b). If the GR-integrals over [a, c] and [c, b] exist, we can define the GR integral of f over [a, b] by taking equation (6) as definition. Similarly, if f is continuous on the interior but singular at both endpoints, and if for some interior point c the GR integrals over [a, c] and [c, b] exist, then we can again take (6) as a definition of the left-hand side. Finally, if we have a function which is continuous [a, b] except for singular points s1, . . . , sn, we can chop up [a, b] into a finite number of sub-intervals on which f has only one singularity (intersperse non-singular points yi with the si’s), and use the analog of equation (6) with one term for each sub-interval to define the left-hand side. Looking over what we’ve just said, we see that we never really needed f to be continuous off the singular points (though that’s most commonly what we see in practice). Our formal definition becomes: Definition 3. Suppose the real-valued function f has the following property: there exist points s0, s1, . . . , sn+1, with a = s0 and b = sn+1, such that f is defined at every point of [a, b] except possibly the si’s, and such that for 0 ≤ i ≤ n, f is GR-integrable over [si, si+1] in the sense of Definition 1. Then we say that f is GR-integrable over [a, b], and define ∫ b a f(t)dt = n∑ i=0 ∫ si+1 si f(t)dt. (7) (Note that we don’t require f to be singular at the si’s; we simply allow it. In general, to apply Definition 1 we’ll have to intersperse nonsingular points between the singular points.) There is a potential problem with this definition. In general, if f is GR-integrable over [a, b], there will be infinitely many choices for the si. For example, if f(x) = [x(1−x)]−1/2 on [0,1], we could choose s1 to be any number strictly between 0 and 1. For the integral over [0, 1] to be well-defined, we need to know that the right-hand side of (6) does not depend on where we put the non-singular si’s. Exercise. 8. Prove that for functions GR-integrable according to Definition 3, ∫ b a f(t)dt is well-defined (i.e. does not depend on the choice of the points si). §2 Integrals over unbounded intervals. Next we want to allow for the possibility of integrating functions over infinite intervals (e.g. [0,∞)). The most intuitive way to do this is the following. Definition 4. Let a ∈ R. We say f is GR-integrable on [a,∞) (or “ ∫∞ a f(t)dt 5 exists”, or “ ∫∞ a f(t)dt converges”) iff (i) for every y > a, f is GR-integrable over [a, y] (in the sense of Definition 3), and (ii) limy→∞ ∫ y a f(t)dt exists. In the GR-integrable case, we define ∫∞ a f(t)dt to be the value of this limit. We define “GR-integrable over (−∞, a]” and “ ∫ a −∞ f(t)dt” similarly. We say f is GR-integrable on (−∞,∞) if it is GR-integrable on (−∞, 0] and [0,∞), in which case we define ∫∞ −∞ f(t)dt = ∫ 0 −∞ f(t)dt+ ∫∞ 0 f(t)dt. Exercises. 9. Prove that, in the definition of integrability over (−∞,∞), the number “0” could have been replaced by any real number without changing the set of functions being called GR-integrable or (in the GR-integrable case) the value of the integral. 10. Let a > 0. Prove that ∫∞ a 1 xp dx exists iff p > 1. (Here p can be any real number.) 11. Determine all values of p for which ∫∞ 0 1 xp dx exists. 12. Let a ∈ R. Assume f is GR-integrable on [a, y] for all y > a. Assume there exists a function g : (a,∞) → R, GR-integrable on [a,∞), such that |f(x)| ≤ g(x), ∀x > a. Prove that f is GR-integrable on [a,∞) and that | ∫∞ a f(x)dx| ≤ ∫∞ a g(x)dx. 13. Let a ∈ R. Assume f is continuous on (a,∞) and that, for some p > 1, the function x 7→ (x− a)pf(x) is bounded on (a,∞). Prove that ∫∞ a f(t)dt exists. 14. Let  > 0. Let P be a polynomial function. Prove that, no matter how small  is or how large the degree of P is, ∫∞ 0 e −xP (x)dx converges. 15. Suppose f is defined on [a,∞) and f(x) ≥ 0 ∀x. Prove that if ∫∞ a f(x)dx exists, then lim infx→∞ f(x) = 0. (Note: previously we defined “lim inf” only for sequences, but you should be able to figure out how to extend the definition to the current situation.) Remark. The statement in exercise 15 would be false if “lim inf” were replaced by “lim”. First, limx→∞ f(x) doesn’t have to exist for ∫∞ a f(x)dx to exist. Second, it is even possible for the integral to converge if there is sequence xn →∞ for which f(xn)→∞. As an example, consider a function f which is zero most places, except for triangular spikes centered at the positive integers. For the spike centered at n, let the base of the spike have width 2−2n and height 2 · 2n, so that the triangle has area 2−n. Then it’s not hard to show that f is GR-integrable on [0,∞) and that the integral equals the (convergent) geometric series ∑∞ 1 2 −n, even though f(n) → ∞. If we drop the restriction that f be nonnegative, it is easy to come up with other examples of counterintuitive phenomena; see exercises 17-19. §3 Change-of-Variables Formula. Often a formal change of variables made to simplify the computation of an integral turns a “proper” integral into an “improper” one, or vice-versa. Sometimes a change of variables turns one “improper” integral into another. One needs to know whether such changes of variable are valid. For simplicity, we will state the theorem only for functions defined on an interval of the form (a, b] or [b,∞) and which satisfy the corresponding 6 parts of Definitions 1 or 4. A more general statement isn’t hard to prove, but is messier to state. Change-of-Variables Theorem. Suppose f is continuous on an interval I, where either (i) I = (a, b] or (ii) I = [b,∞). Let φ : I → R be continuous on I and continuously differentiable on the interior of I. In case (i), suppose that lim t↓a φ(t) = c, where we allow the symbol c to stand for a real number or for ∞. (Thus we assume that either the limit exists, or that limt↓a φ(t) =∞.) Similarly, in case (ii), suppose that lim t→∞ φ(t) = c, where again c may stand for∞. Then in each case ((i) and (ii)) either both of the integrals∫ b a f(φ(t))φ′(t)dt, ∫ φ(b) c f(x)dx (8) exist, or neither does. (If c = ∞ or c ≥ φ(b), see the last remark at the end of these notes.) When the integrals exist, they are equal. Proof. We will write out the proof only for the case in which I = [a, b) and c < φ(b) is a real number; the other cases are similar. Assume the second integral in (8) exists. Note that for u > a, the hypotheses of Corollary 3 on p. 128 of Rosenlicht (the change-of-variables formula for “proper” integrals) are satisfied with U = (u, b). By hypothesis limy↓c ∫ φ(b) y f(x)dx exists and limu↓a φ(u) = c. Hence ∫ φ(b) c f(x)dx = lim y↓c ∫ φ(b) y f(x)dx = lim u↓a ∫ φ(b) φ(u) f(x)dx = lim u↓a ∫ b u f(φ(t))φ′(t)dt. Thus the limit on the extreme right, which is the definition of ∫ b a f(φ(t))φ ′(t)dt, exists and equals the integral on the extreme left. Conversely, suppose that the first integral in (8) exists. Then, as above, we have∫ b a f(φ(t))φ′(t)dt = lim u↓a ∫ b u f(φ(t))φ′(t)dt = lim u↓a ∫ φ(b) φ(u) f(x)dx. Let yn ↓ c. Since limt↓a φ(t) = c, the Intermediate Value Theorem implies that φ maps the interval (a, a + δ] onto (c, φ(a + δ)] for all δ < b − a. Thus there exists a sequence un → a such that yn = φ(un) for all n. Continuing the chain of equalities above, we then have lim u↓a ∫ φ(b) φ(u) f(x)dx = lim n→∞ ∫ φ(b) φ(un) f(x)dx = lim n→∞ ∫ φ(b) yn f(x)dx. Applying Lemma 1′, limy↓c ∫ φ(b) y f(x)dx exists and equals the first integral in (8). 7
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