Measure, Integration & Real Analysis · Problem 1A.7

Measures — Sheldon Axler

Mathematics — Measure Theory · riemann-integral-dyadic-partitions

Problem

Suppose f:[a,b]→Rf : [a, b] \to \mathbf{R} is a bounded function. For n∈Z+n \in \mathbf{Z}^+, let PnP_n denote the partition that divides [a,b][a, b] into 2n2^n intervals of equal size. Prove that L(f,[a,b])=lim⁡n→∞L(f,Pn,[a,b])andU(f,[a,b])=lim⁡n→∞U(f,Pn,[a,b]).L(f, [a, b]) = \lim_{n \to \infty} L(f, P_n, [a, b]) \quad \text{and} \quad U(f, [a, b]) = \lim_{n \to \infty} U(f, P_n, [a, b]).

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