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Let $f,g:[a,b]\to\mathbb{R}$ be two Riemann integrable functions. For each partition $P=\{t_0,...,t_n\}$ of $[a,b]$ take some points in two different ways, that is by choosing two points $\eta_i$ and $\xi_i$ belonging to each $[t_{i-},t_i]$.

show that $$\lim_{|p|\to 0}\Sigma_{i=1}^{n}f(\xi_i)g(\eta_i)(t_i-t_{i-1})=\int_{a}^{b}f(x)g(x)dx.$$

I tried to use the definition of upper and lower integral and as $fg$ is integrable we have equality instead of nonequality. However I got nowhere.

Thanks in advance.

Majid
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1 Answers1

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Hint:$$\sum f(\eta_i)g(\xi_i)\Delta_i=\sum f(\eta_i)(g(\xi_i)-g(\eta_i))\Delta_i+\sum f(\eta_i)g(\eta_i)\Delta_i$$ And notice that $f$ is bounded , $g$ is integrable .

J.Guo
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