Problem 45 - A Course in Calculus and Real Analysis

Question

(Bliss Theorem) Let \(f, g:[a, b] \rightarrow \mathbb{R}\) be any functions. For each \(n \in \mathbb{N}\), consider a partition \(P_{n}:=\left\\{x_{n, 0}, x_{n, 1}, \ldots, x_{n, k_{n}}\right\\}\) of \([a, b]\), and for \(i=\) \(1, \ldots, k_{n}\), let \(s_{n, i}, t_{n, i} \in\left[x_{n, i-1}, x_{n, i}\right]\), and let $$ \widetilde{S}_{n}:=\sum_{i=1}^{k_{n}} f\left(s_{n, i}\right) g\left(t_{n, i}\right)\left(x_{n, i}-x_{n, i-1}\right) . $$ Show that if \(f\) is integrable, \(g\) is continuous, and \(\mu\left(P_{n}\right) \rightarrow 0\), then $$ \tilde{S}_{n} \rightarrow \int_{a}^{b} f(x) g(x) d x $$ (Hint: If \(\mathcal{T}_{n}:=\left\\{s_{n, i}: i=1, \ldots, k_{n}\right\\}\), then \(S\left(P_{n}, \mathcal{T}_{n}, f g\right) \rightarrow \int_{a}^{b} f(x) g(x) d x\) Also, \(g\) is uniformly continuous.)

Step-by-Step Solution

Answer

In summary, we showed that if \(f\) is integrable, \(g\) is continuous, and the mesh size \(\mu(P_n)\) tends to 0, then \(\widetilde{S}_{n} \rightarrow \int_{a}^{b} f(x)g(x) dx\) by connecting the convergence of the sum \(\widetilde{S}_{n}\) to the convergence of the Riemann sum of the product function \(fg\). We used the fact that \(g\) is uniformly continuous, bound the difference between the sum and the integral, and then applied the squeeze theorem.

Step 1: Identify the limit of the Riemann sum

Given that \(\mathcal{T}_{n}:=\left\{s_{n, i}: i=1, \ldots, k_{n}\right\}\), we have \(S(P_{n}, \mathcal{T}_{n}, fg) \rightarrow \int_{a}^{b} f(x)g(x)dx\). This is a standard Riemann sum of the function \((fg)\) on the partition \(P_n\) with corresponding points \(s_{n, i}\).

Step 2: Show g is uniformly continuous

Since \(g\) is continuous on \([a, b]\), which is a compact set, it follows that \(g\) is also uniformly continuous on \([a, b]\).

Step 3: Find an upper bound for the difference between the sum and integral

Now, let's find an upper bound for the difference between \(\widetilde{S}_{n}\) and the integral \(\int_{a}^{b} f(x)g(x) dx\). We can write: \[ | \widetilde{S}_{n} - \int_{a}^{b} f(x)g(x) dx | \leq \sum_{i=1}^{k_{n}} \left| f(s_{n, i})g(t_{n, i}) - f(s_{n, i})g(s_{n, i}) \right| (x_{n, i} - x_{n, i-1}) \] Using the fact that \(g\) is uniformly continuous, there exists a \(\delta > 0\) such that for all \(x, y \in [a, b]\), if \(|x - y| < \delta\), then \(|g(x) - g(y)| < \epsilon/(\|f\|_{\infty}(b-a))\), where \(\|f\|_{\infty} = \sup_{x\in[a,b]} |f(x)|\). Since \(\mu(P_n) \rightarrow 0\), there exists an \(N \in \mathbb{N}\) such that for \(n \geq N\), \(\mu(P_n) < \delta\). For these \(n\), we have: \[ |g(t_{n, i}) - g(s_{n, i})| < \frac{\epsilon}{\|f\|_{\infty}(b-a)} \] which implies, for \(n \geq N\): \[ | \widetilde{S}_{n} - \int_{a}^{b} f(x)g(x) dx | \leq \sum_{i=1}^{k_{n}} | f(s_{n, i})| |g(t_{n, i}) - g(s_{n, i})| (x_{n, i} - x_{n, i-1}) \leq \sum_{i=1}^{k_{n}} \|f\|_{\infty} \frac{\epsilon}{\|f\|_{\infty}(b-a)} (x_{n, i} - x_{n, i-1}) = \epsilon \]

Step 4: Apply the squeeze theorem

Now that we have an upper bound for the difference between \(\widetilde{S}_{n}\) and the integral \(\int_{a}^{b} f(x)g(x) dx\) for \(n \geq N\), we can use the squeeze theorem to show that \(\widetilde{S}_{n} \rightarrow \int_{a}^{b} f(x)g(x) dx\). Since the difference \(|\widetilde{S}_{n} - \int_{a}^{b} f(x)g(x) dx|\) is less than \(\epsilon\) for all \(\epsilon > 0\), it follows that the difference goes to zero as \(n \rightarrow \infty\), and hence: \[ \lim_{n \to \infty} \widetilde{S}_{n} = \int_{a}^{b} f(x)g(x) dx \] Thus, we have shown that \(\widetilde{S}_{n} \rightarrow \int_{a}^{b} f(x)g(x) dx\) as required.

Key Concepts

Riemann Sum

Understanding Riemann sums is crucial in the world of calculus, particularly when dealing with the fundamental concept of integration. A Riemann sum is a technique for approximating the total area under a curve, which is known as the integral of a function over an interval. To visualize this, imagine dividing the interval on the x-axis into smaller sub-intervals, and over each, constructing a rectangle whose height is determined by the function value at a certain point within the sub-interval.

The sum of the areas of these rectangles gives us a Riemann sum. When the number of rectangles increases, and the width of each sub-interval decreases, the Riemann sum gets closer to the exact area, namely the integral of the function. This is precisely the process highlighted in the Bliss Theorem exercise where the partitioning of the interval \[a, b\] and taking the sum over all sub-intervals was used to approximate the integral \(\int_{a}^{b} f(x)g(x) dx\).

Uniform Continuity

Uniform continuity is a stronger form of continuity that applies to functions over an interval. While a function being continuous at a point means that small changes in the input result in small changes in the output at that point, uniform continuity takes this a step further.

A function is uniformly continuous on an interval if, regardless of where you are on the interval, any small change in input yields a small change in output within a specific bound. This concept is pivotal in the provided exercise, as the uniformly continuous nature of \(g\) ensures that for any two points \(x\) and \(y\) in the interval \([a, b]\), the function values \(g(x)\) and \(g(y)\) will be close provided that \(x\) and \(y\) are close. This property facilitates the estimation of the function's integral using Riemann sums, by boundedly approximating the values of \(g\) within sub-intervals of the partition \(P_n\).

Squeeze Theorem

The squeeze theorem, sometimes referred to as the sandwich theorem, is a fundamental result that lets us determine the limit of a function. The theorem is invoked when you have a function 'squeezed' between two other functions whose limits are known and equal.

In the context of the Bliss Theorem exercise, the squeeze theorem is applied in the final step of the proof. We show that the difference in the area approximated by the Riemann sum \(\widetilde{S}_{n}\) and the true area under the curve given by \(\int_{a}^{b} f(x)g(x) dx\) can be made arbitrarily small. Since this difference can be squeezed down below any \(\epsilon > 0\), effectively becoming zero, we conclude through the squeeze theorem that the limit of \(\widetilde{S}_{n}\) as \(n\) approaches infinity is indeed \(\int_{a}^{b} f(x)g(x) dx\).

Integrable Functions

An integrable function is one over which the integral, or the area under the curve, can be properly defined and is finite. The concept of integrability isn't just about whether we can 'find the area', but also how it behaves within the formal structure of the integral calculus. For a function to be integrable on an interval, it must largely be continuous, or have a finite number of discontinuities.

In our Bliss Theorem exercise, the function \(f\) is given as an integrable function, which implies that we can evaluate its integral over the interval \([a, b]\). This feature of \(f\) plays a vital role in establishing the convergence of the Riemann sum to the true integral of the product \(f(x)g(x)\). An integrable function aids in ensuring the limit of the Riemann sums represents the actual area under the curve with increasing accuracy.