Measure, Integration & Real Analysis · Problem 1A.8

Measures — Sheldon Axler

Mathematics — Measure Theory · riemann-integral-limit-of-sums

Problem

Suppose f:[a,b]→Rf : [a, b] \to \mathbf{R} is Riemann integrable. Prove that ∫abf=lim⁡n→∞b−an∑j=1nf ⁣(a+j(b−a)n).\int_a^b f = \lim_{n \to \infty} \frac{b - a}{n} \sum_{j=1}^n f\!\left(a + \frac{j(b-a)}{n}\right).

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