1.1. Prologue: The problem of measure 13 is independent of the choice of partition used to demonstrate the piecewise constant nature of f. We will denote this quantity by p.c. b a f(x) dx, and refer to it as the piecewise constant integral of f on [a, b]. Exercise 1.1.21 (Basic properties of the piecewise constant integral). Let [a, b] be an interval, and let f, g : [a, b] → R be piecewise constant functions. Establish the following statements: (1) (Linearity) For any real number c, cf and f + g are piecewise con- stant, with p.c. b a cf(x) dx = c p.c. b a f(x) dx and p.c. b a f(x) + g(x) dx = p.c. b a f(x) dx + p.c. b a g(x) dx. (2) (Monotonicity) If f ≤ g pointwise (i.e. f(x) ≤ g(x) for all x ∈ [a, b]), then p.c. b a f(x) dx ≤ p.c. b a g(x) dx. (3) (Indicator) If E is an elementary subset of [a, b], then the in- dicator function 1E : [a, b] → R (defined by setting 1E(x) := 1 when x ∈ E and 1E(x) := 0 otherwise) is piecewise constant, and p.c. b a 1E(x) dx = m(E). Definition 1.1.6 (Darboux integral). Let [a, b] be an interval, and f : [a, b]→ R be a bounded function. The lower Darboux integral bf(x) a dx of f on [a, b] is defined as b a f(x) dx := sup g≤f, piecewise constant p.c. b a g(x) dx, where g ranges over all piecewise constant functions that are pointwise bounded above by f. (The hypothesis that f is bounded ensures that the supremum is over a non-empty set.) Similarly, we define the upper Darboux integral b a f(x) dx of f on [a, b] by the formula b a f(x) dx := inf h≥f, piecewise constant p.c. b a h(x) dx. Clearly, b a f(x) dx ≤ b a f(x) dx. If these two quantities are equal, we say that f is Darboux integrable, and refer to this quantity as the Darboux integral of f on [a, b]. Note that the upper and lower Darboux integrals are related by the reflection identity b a − f(x) dx = − b a f(x) dx. Exercise 1.1.22. Let [a, b] be an interval, and f : [a, b] → R be a bounded function. Show that f is Riemann integrable if and only if it is Darboux

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2011 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.