Question

Suppose \(\alpha\) increases on \([a, b], a \leq x_{0} \leq b, \alpha\) is continuous at \(x_{0}, f\left(x_{0}\right)=1\), and \(f(x)=0\) if \(x \neq x_{0} .\) Prove that \(f \in \Omega(\alpha)\) and that \(\int f d \alpha=0 .\)

Short answer

Question: Prove that the given function \(f\) belongs to the class \(\Omega(\alpha)\) and evaluate \(\int f \: d\alpha\). Answer: \(f \in \Omega(\alpha)\) and \(\int f \: d\alpha = 0\).

Step-by-step solution

Understanding the function f and α

We have a function \(f\) defined on the interval \([a, b]\) such that \(f(x_0) = 1\) and \(f(x) = 0\) if \(x \neq x_0\). The function \(\alpha\) is increasing on the interval \([a, b]\) and is continuous at \(x_0\).

Showing L(P, f, α) = 0

We want to show that for any partition \(P = \{x_0, x_1, \dots, x_n\}\) of \([a, b]\), the lower Riemann-Stieltjes sum \(L(P, f, \alpha) = 0\). Let's recall the definition of the lower Riemann-Stieltjes sum: \(L(P, f, \alpha) = \sum_{i=1}^{n} m_i(\alpha(x_i) - \alpha(x_{i-1}))\) where \(m_i = \inf_{x \in [x_{i-1}, x_i]} f(x)\). Since \(f(x) = 0\) for all \(x \neq x_0\), we know that \(m_i = 0\) for all \(i\) except possibly when the interval \([x_{i-1}, x_i]\) contains \(x_0\). When the interval \([x_{i-1}, x_i]\) contains \(x_0\), the infimum of \(f(x)\) in that interval is also 0, because \(f(x)\) is still 0 for all other values. Therefore, we have that \(m_i = 0\) for all \(i\), so \(L(P, f, \alpha) = 0\) in every case.

Showing f is in Ω(α)

Given that \(L(P, f, \alpha) = 0\) for any partition \(P\) of \([a, b]\), we can say the following: \(\underline{\int_{a}^{b}} f \: d\alpha = \sup_{P}{L(P, f, \alpha)} = 0\) Since the lower Riemann–Stieltjes integral exists, it can be concluded that \(f \in \Omega(\alpha)\).

Evaluating the integral

Having shown that \(f \in \Omega(\alpha)\), we can now evaluate \(\int f \: d\alpha\): \(\int f \: d\alpha = \sup_{P}{L(P, f, \alpha)} = 0\) Therefore, the integral of \(f \: d\alpha\) is equal to 0.

Key concepts

Function Continuity

In the given exercise, one key concept is the continuity of the function \(\alpha\). A function is continuous at a point if there are no sudden jumps or breaks at that point. Mathematically, this means that \(\lim_{x \to x_0} \alpha(x) = \alpha(x_0)\).

Function continuity ensures that the behavior of the function is predictable. Having continuous functions in calculus is crucial because it allows us to apply certain theorems and procedures, like the Riemann-Stieltjes integral, without worrying about erratic behavior.

In this context, the continuity of \(\alpha\) at \(x_0\) is significant because it guarantees that any change over the interval \([a, b]\) includes this point smoothly. Since \(\alpha\) is increasing and continuous at \(x_0\), there are no disruptions at this point, which is pivotal for defining and evaluating the integral in question.

Increasing Functions

An increasing function is one where, as the input (or \(x\)-value) increases, the output (or \(\alpha(x)\)) does not decrease. Specifically, for any two points \(x_1\) and \(x_2\) in the domain, if \(x_1 < x_2\), then \(\alpha(x_1) \leq \alpha(x_2)\).

This property is important in the context of the Riemann-Stieltjes integral because it influences how the integral is calculated. When a function is increasing, the differences in the integral formula \( (\alpha(x_i) - \alpha(x_{i-1})) \) are always non-negative.

Since \(\alpha\) is increasing on \([a, b]\), it implies a non-decreasing overall trend across the interval. This characteristic aligns with the exercise's requirement to show that \(f \in \Omega(\alpha)\) and further assures that the calculations involving \(\alpha\) are mathematically valid, given the nature of such input functions.

Integral Calculus

Integral calculus deals with the concept of integration, which is essentially the process of calculating the area under a curve. In this exercise, we use the concept of the Riemann-Stieltjes integral, which is a generalization of the Riemann integral.

The Riemann-Stieltjes integral \( \int f \ d \alpha \) is used when you want to integrate a function \(f(x)\) with respect to another function \(\alpha(x)\). This allows for more flexibility than standard integration and is useful when factoring in functions like \(\alpha\) that may add specific properties or behavior to the integral, such as monotonicity or discontinuities.

To show that the Riemann-Stieltjes integral is 0, it involves analyzing the lower sum, and here, we needed to establish that for any partition \(P\), the lower sum \(L(P, f, \alpha) = 0\). Since \(f(x) = 0\) almost everywhere, except at one point, and considering \(\alpha\) is continuous and increasing, it leads to the integral being zero.

This profound concept shows the intersection of calculus with function properties, allowing us to process complex integrals by acknowledging the roles of continuity, increments, and specific function definitions.