# Download A (terse) introduction to Lebesgue integration by John Franks PDF By John Franks

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Show there exists a family {fn }n∈N of step functions such that for each n and each x ∈ [a, b] the value of fn (x) is either 0 or 1 and yet the function deﬁned by g(x) = max{fn (x)}n∈N is not Riemann integrable. (4) Prove that if f and g are bounded Riemann integrable functions on an interval [a, b], then so is f g. In particular, if r ∈ R, then rf is a bounded Riemann integrable function on [a, b]. 1. Introduction In the previous section we studied two deﬁnitions of integration that were based on two important facts: (1) There is only one obvious way to deﬁne the integral of step functions assuming we want it to satisfy certain basic properties, and (2) these properties force the definition for the integral for more general functions which are uniformly approximated by step functions (regulated integral) or squeezed between step functions whose integrals are arbitrarily close (Riemann integral).

3. A bounded function f : [a, b] → R is Riemann integrable if and only if, for every ε > 0 there are step functions v0 and u0 such that v0 (x) ≤ f (x) ≤ u0 (x) for all x ∈ [a, b] and b a b u0 (x) dx − v0 (x) dx ≤ ε. a Proof. Suppose the functions v0 ∈ L(f ) and u0 ∈ U(f ) have integrals within ε of each other. 1. This implies b u∈U(f ) b u(x) dx − sup inf v∈L(f ) a a v(x) dx ≤ ε. Since this is true for all ε > 0, we conclude that f is Riemann integrable. Conversely, if f is Riemann integrable, then from the properties of the inﬁmum there exists a step function u0 ∈ U(f ) such that b a b u0 (x) dx < u(x) dx + inf u∈U(f ) Thus, a b b u0 (x) dx − a ε = 2 f (x) dx < a b a ε f (x) dx + .

N=1 • Regularity: If A ∈ M(I), then µ(A) = inf{µ(U ) | U is open and A ⊂ U }. • Increasing sequences: If An ∈ M(I) for n ∈ N satisfy An ⊂ An+1 , then ∞ µ An = lim µ(An ). n=1 n→∞ • Decreasing sequences: If An ∈ M(I) for n ∈ N satisfy An ⊃ An+1 , then ∞ An = lim µ(An ). 1. (The Cantor middle third set). Recursively deﬁne a nested sequence {Jn }∞ n=0 of closed subsets of I = [0, 1]. Each Jn consists of a ﬁnite union of closed intervals. We deﬁne J0 to be I and let Jn be the union of the closed intervals obtained by deleting the open middle third interval from each of the intervals in Jn−1 .