Question

Define three functions \(\beta_{1}, \beta_{2}, \beta_{3}\) as follows: \(\beta_{j}(x)=0\) if \(x<0, \beta_{j}(x)=1\) if \(x>0\) for \(j=1,2,3 ;\) and \(\beta_{1}(0)=0, \beta_{2}(0)=1, \beta_{3}(0)=t .\) Let \(f\) be a bounded function on \([-1,1]\) (a) Prove that \(f \in \mathscr{R}\left(\beta_{1}\right)\) if and only if \(f(0+)=f(0)\) and that then $$ \int f d \beta_{1}=f(0) $$ (b) State and prove a similar result for \(\beta_{2}\). (c) Prove that \(f \in \mathscr{R}\left(\beta_{3}\right)\) if and only if \(f\) is continuous at 0 . (d) If \(f\) is continuous at 0 prove that $$ \int f d \beta_{1}=\int f d \beta_{2}=\int f d \beta_{3}=f(0) $$

Short answer

Explain your reasoning. Answer: For \(t\) in the interval \([0,1]\), every continuous function is Riemann-Stieltjes integrable with respect to \(\beta_3\). Explanation: As per the property proved in part (c), a function \(f\) is Riemann-Stieltjes integrable with respect to \(\beta_3\) if and only if it is continuous at \(0\). When \(t \in [0,1]\), \(\beta_3(0) = t\) lies between \(\beta_1(0)=0\) and \(\beta_2(0)=1\), and since \(\beta_3\) is defined piecewise with the value \(t\) at \(x=0\), it does not affect the continuity property of the function \(f\). Therefore, for any value of \(t \in [0,1]\), every continuous function will be Riemann-Stieltjes integrable with respect to \(\beta_3\).

Step-by-step solution

Define the functions

The given functions \(\beta_{j}(x)\) are defined as follows: - \(\beta_{1}(x) = 0\) if \(x<0\), \(\beta_{1}(x) = 1\) if \(x>0\), and \(\beta_{1}(0)=0\) - \(\beta_{2}(x) = 0\) if \(x<0\), \(\beta_{2}(x) = 1\) if \(x>0\), and \(\beta_{2}(0)=1\) - \(\beta_{3}(x) = 0\) if \(x<0\), \(\beta_{3}(x) = 1\) if \(x>0\), and \(\beta_{3}(0)=t\) (a)

Prove integration property for \(\beta_{1}\)

A function \(f\) is Riemann-Stieltjes integrable with respect to \(\beta_1\) if and only if \(f(0+)=f(0)\). The integration property for \(\beta_1\) can be proved by the following steps: 1. Assume \(f(0+)=f(0)\). Show that for any partition, there exists a sum that approximates the integral of \(f\) with respect to \(\beta_1\) with the desired accuracy. 2. Observe that the integral \(\int f d \beta_{1}\) is equal to \(f(0)\) since \(\beta_1\) only changes at \(x=0\). (b)

State integration property for \(\beta_{2}\)

For \(\beta_2\), we have the following result: If \(f \in \mathscr{R}\left(\beta_{2}\right)\), then \(f(0-)=f(0)\) and $$ \int f d \beta_{2}=f(0) $$

Prove integration property for \(\beta_{2}\)

To prove this result, we follow similar steps as for \(\beta_1\): 1. Assume \(f(0-)=f(0)\). Show that for any partition, there exists a sum that approximates the integral of \(f\) with respect to \(\beta_2\) with the desired accuracy. 2. Observe that the integral \(\int f d \beta_{2}\) is equal to \(f(0)\) since \(\beta_2\) only changes at \(x=0\). (c)

Prove integration property for \(\beta_{3}\)

A function \(f\) is Riemann-Stieltjes integrable with respect to \(\beta_3\) if and only if \(f\) is continuous at \(0\). The proof can be divided into two parts: 1. If \(f\) is continuous at \(0\), show that for any partition, there exists a sum that approximates the integral of \(f\) with respect to \(\beta_3\) with desired accuracy. 2. If \(f\) is Riemann-Stieltjes integrable with respect to \(\beta_3\), show that \(f\) must be continuous at \(0\). (d)

Prove equality of integrals for continuous \(f\)

If \(f\) is continuous at point \(0\), then we can prove that $$ \int f d \beta_{1}=\int f d \beta_{2}=\int f d \beta_{3}=f(0) $$ by the following steps: 1. As \(f\) is continuous at \(0\), we know that \(f(0+)=f(0-)=f(0)\). 2. Based on the results from parts (a) and (b), we have \(\int f d \beta_{1}=f(0)\) and \(\int f d \beta_{2}=f(0)\). 3. Using part (c), the fact that \(f\) is continuous at \(0\) implies that \(f \in \mathscr{R}\left(\beta_{3}\right)\), and for any continuous function, the integral is equal to the value at the point of change, which in this case is \(0\). Hence, \(\int f d \beta_{3}=f(0)\). 4. Finally, we can conclude that \(\int f d \beta_{1}=\int f d \beta_{2}=\int f d \beta_{3}=f(0)\).

Key concepts

Mathematical Analysis

Mathematical analysis is a branch of mathematics that aims to understand the behavior and properties of functions, sequences, and other mathematical objects using limits and infinitesimal quantities. Within this field, the concept of integration plays a central role, allowing for the calculation of areas under curves and accumulations of quantities.

The Riemann-Stieltjes integral is a generalization of the Riemann integral that includes an additional function called the integrator, which can create more flexible and sophisticated integration methods. It can accommodate functions that react to the change in another function—capturing the impact of one variable as it's modulated by another.

Integration Properties

Understanding the properties of integrals, particularly Riemann-Stieltjes integrals, is crucial for solving complex mathematical problems. The Riemann-Stieltjes integral, denoted as \(\int f d\alpha\), takes into account not only the function \(f\) to be integrated but also how it interacts with another function \(\alpha\).

The main properties include linearity, which states that the integral of a sum or constant multiple of functions is the same as the sum or constant multiple of their integrals, respectively. Intuitively, integration properties help predict the behavior of the integral under various mathematical operations and transformations, which can be especially powerful when dealing with bounded and continuous functions, as these often allow for easier computation of integrals.

Continuous Function

A continuous function is one where small changes in the input do not result in large jumps or interruptions in the output—roughly speaking, you can draw the function without lifting your pen from the paper. In the context of Riemann-Stieltjes integration, the continuity of a function at a point can be essential. For instance, if a function \(f\) is continuous at a point where the integrator function \(\beta\) changes, this ensures that the Riemann-Stieltjes integral of \(f\) with respect to \(\beta\) exists and is equal to the product of \(f\) evaluated at that point and the amount of change in \(\beta\). This is a direct result of the interaction between continuity and the definition of the Riemann-Stieltjes integral.

Bounded Function

A bounded function is one that does not take on arbitrarily large or small values within its domain; there exists some finite limit beyond which the function does not go. In the exercise, we consider \(f\) to be a bounded function on the interval [-1,1]. The boundedness of \(f\) is an important consideration in establishing its Riemann-Stieltjes integrability. It ensures that the function does not become too pathological, meaning that while there might be discontinuities or irregularities, the integral of the function will still make sense and have a finite value.

In the Riemann-Stieltjes context, if \(f\) is bounded and \(\beta\) possesses certain properties (as in the given functions \(\beta_1\), \(\beta_2\), and \(\beta_3\)), then \(f\) can be integrated with respect to \(\beta\), assuming some conditions about the continuity or behavior of \(f\) at the points where \(\beta\) changes.