Question

Let \(f\) be continuous and \(g\) be of bounded variation (Exercise 3.2.7) on \([a, b]\). (a) Show that if \(\epsilon>0,\) there is a \(\delta>0\) such that \(\left|\sigma-\sigma^{\prime}\right|<\epsilon / 2\) if \(\sigma\) and \(\sigma^{\prime}\) are Riemann-Stieltjes sums of \(f\) with respect to \(g\) over partitions \(P\) and \(P^{\prime}\) of \([a, b],\) where \(P^{\prime}\) is a refinement of \(P\) and \(\|P\|<\delta\). HINT: Use Theorem \(2.2 .12 .\) (b) Let \(\delta\) be as chosen in (a). Suppose that \(\sigma_{1}\) and \(\sigma_{2}\) are Riemann-Stieltjes sums of \(f\) with respect to \(g\) over any partitions \(P_{1}\) and \(P_{2}\) of \([a, b]\) with norm less than \(\delta\). Show that \(\left|\sigma_{1}-\sigma_{2}\right|<\epsilon\) (c) If \(\delta>0,\) let \(L(\delta)\) be the supremum of all Riemann-Stieltjes sums of \(f\) with respect to \(g\) over partitions of \([a, b]\) with norms less than \(\delta\). Show that \(L(\delta)\) is finite. Then show that \(L=\lim _{\delta \rightarrow 0+} L(\delta)\) exists. HINT: Use Theorem 2.1.9. (d) Show that \(\int_{a}^{b} f(x) d g(x)=L\).

Short answer

In this exercise, we are given a continuous function f and a function g of bounded variation on the interval [a, b]. Our goal is to prove that the Riemann-Stieltjes integral of f with respect to g exists and is equal to a certain limit L. (a) We showed that for any \(\epsilon > 0\), there exists a \(\delta > 0\) such that for any two Riemann-Stieltjes sums over partitions with norm less than \(\delta\), the difference between the sums is less than \(\epsilon / 2\). (b) We used the result from part (a) and the triangle inequality to show that for any two Riemann-Stieltjes sums over partitions with norm less than \(\delta\), the difference between the sums is still less than \(\epsilon\). (c) We showed that there exists a limit L for the supremum of Riemann-Stieltjes sums over partitions with norms less than \(\delta\) as \(\delta\) approaches 0, by using the fact that g is of bounded variation. (d) Finally, we used the results from the previous parts and the triangle inequality to show that for any Riemann-Stieltjes sum over a partition with norm less than \(\delta\), its difference from the limit L is less than \(\epsilon\). This means that the Riemann-Stieltjes integral of f with respect to g exists and is equal to L.

Step-by-step solution

Part (a)

Since the function f is continuous on the interval [a, b], according to Theorem 2.2.12, given any \(\epsilon > 0\), there exists a \(\delta > 0\) such that for any partition P of [a, b] with \(\|P\| < \delta\) and any choice of points \(t_i\) in \([x_{i-1}, x_i]\), we have: $$\left|\sum_{i=1}^n f(t_i)(x_i - x_{i-1}) - \sum_{i=1}^n f(t_i^{'}) (x_i - x_{i-1}) \right| < \frac{\epsilon}{2}$$ Consider \(\sigma = \sum_{i=1}^n f(t_i)\Delta g_i\) and \(\sigma^{'} = \sum_{i=1}^n f(t_i^{'})\Delta g_i\) to be Riemann-Stieltjes sums of f with respect to g for partitions P and \(P^{'}\) respectively, where \(P^{'}\) is a refinement of P. Now we have: $$\left|\sigma - \sigma^{'}\right| = \left|\sum_{i=1}^n f(t_i)\Delta g_i - \sum_{i=1}^n f(t_i^{'}) \Delta g_i \right| \le \sum_{i=1}^n \left|f(t_i)-f(t_i^{'})\right|\Delta g_i$$ Using Theorem \(2.2.12\), we get \(\left|\sigma-\sigma^{'}\right| < \epsilon/2\) for any Riemann-Stieltjes sums \(\sigma\) and \(\sigma^{'}\) with \(\|P\|<\delta\).

Part (b)

Let \(\delta\) be as chosen in part (a). Let \(\sigma_{1}\) and \(\sigma_{2}\) be Riemann-Stieltjes sums of f with respect to g over partitions \(P_{1}\) and \(P_{2}\) of \([a, b]\) with norm less than \(\delta\). Let \(P\) be a common refinement of \(P_{1}\) and \(P_{2}\). Then both \(P_{1}\) and \(P_{2}\) are refinements of \(P\). According to part (a), we have $$\left|\sigma_1 - \sigma_P\right| < \epsilon / 2 \quad \text{and} \quad \left|\sigma_2 - \sigma_P\right| < \epsilon / 2$$ Using the triangle inequality, we get $$\left|\sigma_1 - \sigma_2\right| \le \left|\sigma_1 - \sigma_P\right| + \left|\sigma_2 - \sigma_P\right| < \epsilon / 2 + \epsilon / 2 = \epsilon$$

Part (c)

If \(\delta > 0\), let \(L(\delta)\) be the supremum of all Riemann-Stieltjes sums of f with respect to g over partitions of \([a, b]\) with norms less than \(\delta\). Since g is of bounded variation on \([a, b]\), there exists a constant M such that \(Var(g, [a, b]) \le M\). Therefore, \(L(\delta)\) is finite because the maximum value of a Riemann-Stieltjes sum is achieved when f is constant and equals its maximum value on the interval, and the sum is bounded by M. Now, we want to show that \(L = \lim_{\delta \rightarrow 0+} L(\delta)\) exists. According to Theorem 2.1.9, there exists a sequence of partitions \(\{P_k\}\) such that \(\|P_k\| \rightarrow 0\) as \(k \rightarrow \infty\) and \(\lim_{k \rightarrow \infty} \sigma(P_k) = L\), where \(\sigma(P_k)\) is a Riemann-Stieltjes sum over partition \(P_k\). Therefore, \(L = \lim_{\delta \rightarrow 0+} L(\delta)\) exists.

Part (d)

Let \(L\) be as chosen in part (c). We want to show that \(\int_{a}^{b} f(x)d g(x) = L\). According to part (b), given any \(\epsilon > 0\), there exists a \(\delta > 0\) such that for any partitions \(P_1\) and \(P_2\) of [a, b] with norms less than \(\delta\), we have: $$\left|\sigma_1 - \sigma_2\right| < \epsilon$$ where \(\sigma_1\) and \(\sigma_2\) are Riemann-Stieltjes sums of f with respect to g over partitions \(P_1\) and \(P_2\), respectively. Since \(L = \lim_{\delta \rightarrow 0+} L(\delta)\), for the chosen \(\delta\), we have \(\left|L - L(\delta)\right| < \epsilon / 2\). Also, for any \(\sigma\) with respect to a partition with norm less than \(\delta\), we have \(\left|\sigma - L(\delta)\right| < \epsilon / 2\). Therefore, using the triangle inequality: $$\left|\sigma - L\right| \le \left|\sigma - L(\delta)\right| + \left|L(\delta) - L\right| < \epsilon$$ Since the inequality holds for every \(\epsilon > 0\), we can conclude that \(\int_{a}^{b} f(x)d g(x) = L\).

Key concepts

Bounded Variation

Understanding the term bounded variation is essential when diving into the intricacies of the Riemann-Stieltjes integral. Suppose we have a function g defined on an interval [a, b]. We say that g is of bounded variation if there is a ceiling to the total amount by which g can climb or descend as we travel from a to b. Conceptually, this means that if you imagine hiking along the graph of g, the total distance you climb up plus the total distance you descend is bounded, no matter the path you take.

To be more precise, for each partition P of the interval [a, b], consisting of points a=x_0 < x_1 < ... < x_n=b, we look at the sum of the absolute differences |g(x_i) - g(x_{i-1})|. The function g has bounded variation if the supremum of these sums over all possible partitions of [a, b] is capped by some finite number. If the function doesn't have this property, it could have infinite variation, like some fractal curves. However, when dealing with Riemann-Stieltjes integrals, bounded variation of the g function ensures that the integral's value won't be infinitely large due to excessive 'wiggling' of the function.

Continuous Function

When we say that a function f is continuous on an interval [a, b], we're imparting that f behaves nicely; there are no sudden jumps or breaks. Mathematically put, for any point x in the interval, and for any small ϵ>0, there exists a δ>0 such that for all points y within δ of x, the value of f(y) is within ϵ of f(x).

This concept plays an essential role in the Riemann-Stieltjes integral, as the continuity of f ensures small variations in the points chosen for evaluation within any subinterval of a partition will not drastically change the value of the integral. This predictability and 'smoothness' of continuous functions make them particularly manageable and intuitive for integration.

Partition of an Interval

A partition of an interval [a, b] is like dividing a stick into various lengths where the combined length of the pieces makes up the entire stick. Formally, we define a partition P as a finite sequence of increasing numbers starting at a and ending at b, i.e., a = x_0 < x_1 < ... < x_n = b. Each of these intervals [x_{i-1}, x_i] is called a subinterval of the partition.

When we compute Riemann-Stieltjes sums, we use these partitions to break down the interval over which we are integrating into smaller, more digestible pieces. By evaluating the function f at some point within each subinterval and then adding up the product of these values with the respective function g's increments over the subintervals, we obtain an approximation of the integral. As we refine the partition, making the pieces smaller, our approximation should, theoretically, get closer to the true value of the integral.

Supremum of Riemann-Stieltjes Sums

When we talk about the supremum of Riemann-Stieltjes sums over partitions of an interval with norms less than δ, we are looking for the greatest 'achievable' sum without actually going over all possible sums. Imagine multiple people shooting arrows at a target, where each arrow is a different Riemann-Stieltjes sum that comes as close as possible to the bullseye without going over it. The supremum is like the arrow that comes closest to the target.

In practice, to find the supremum, we would evaluate the Riemann-Stieltjes sums over various partitions of [a, b] with finer and finer subintervals (each subinterval being smaller than δ) and choose the largest of these sums. If L(δ) denotes this supremum for a particular δ, we can imagine gradually shrinking δ to zero and looking at the behaviour of the corresponding supremums; this gives us a limit that reflects the actual value of the integral of f with respect to g, as we remove the rough edges of the approximation and zero in on the precise value.