Let $F: {\bf R} \rightarrow {\bf R}$ be monotone non-decreasing, let $[a,b]$ be a compact interval, and let $f: [a,b] \rightarrow {\bf R}$ be continuous. Suppose that $F$ is continuous at the endpoints $a, b$ of the interval. Show that for every $\varepsilon > 0$ there exists $\delta > 0$ such that $$\displaystyle |\sum_{i=1}^n f(t^*_i) (F(t_i) - F(t_{i-1})) - \int_{[a,b]} f\ dF| \leq \varepsilon$$ whenever $a = t_0 < t_1 < \ldots < t_n = b$ and $t^*_i \in [t_{i-1},t_i]$ for $1 \leq i \leq n$ are such that $\sup_{1 \leq i \leq n} |t_i - t_{i-1}| \leq \delta$. In the language of the Riemann-Stieltjes integral, this result asserts that the Lebesgue-Stieltjes integral extends the Riemann-Stieltjes integral.

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An Introduction to Measure Theory

Exercise 1.7.16 (Connection with Riemann-Stieltjes integral)

Let $F : R \to R$ be monotone non-decreasing, let $[a, b]$ be a compact interval, and let $f : [a, b] \to R$ be continuous. Suppose that $F$ is continuous at the endpoints $a, b$ of the interval. Show that for every $𝜀 > 0$ there exists $δ > 0$ such that

| \sum_{i = 1}^{n} f (t_{i}^{∗}) (F (t_{i}) - F (t_{i - 1})) - \int_{[a, b]} f dF | \leq 𝜀

whenever $a = t_{0} < t_{1} < \dots < t_{n} = b$ and $t_{i}^{∗} \in [t_{i - 1}, t_{i}]$ for $1 \leq i \leq n$ are such that $\sup_{1 \leq i \leq n} | t_{i} - t_{i - 1} | \leq δ$ . In the language of the Riemann-Stieltjes integral, this result asserts that the Lebesgue-Stieltjes integral extends the Riemann-Stieltjes integral.

Exercise 1.7.16 (Connection with Riemann-Stieltjes integral)

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