Question

Suppose that \(-\infty

Short answer

In this problem, we were given a function $g(x)$ that is defined differently on different intervals of its domain, and a continuous function $f(x)$. The goal was to show that the integral $\int_{a}^{b} f(x) d g(x)$ exists and to find its value. We achieved this by breaking down the integral into sub-intervals based on the behavior of $g(x)$ and computing the integrals separately. Finally, we found the value of the integral by adding these individual integrals as $g_1 [f(d-) - f(a+)] + g_2 [f(b-) - f(d+)]$.

Step-by-step solution

Understanding \(g(x)\) and the intervals

We are given function \(g(x)\) defined as: $$ g(x)=\left\\{\begin{array}{ll} g_{1}, & a<x<d \\ g_{2}, & d<x<b \end{array}\right. $$ The key thing to note here is that \(g(x)\) is constantly \(g_1\) for all \(x\) between \(a\) and \(d\), and constantly \(g_2\) for all \(x\) between \(d\) and \(b\). This means that \(g(x)\) will only have jumps at points \(a, b,\) and \(d\).

Breaking the integral into sub-intervals

Now that we have identified the points at which \(g(x)\) has jumps, we can consider the integral of \(f(x)\) with respect to \(g(x)\) separately for each sub-interval. We will compute the following integrals and then sum them up: $$ \int_a^d f(x)dg(x) + \int_d^b f(x)dg(x) $$

Computing the first integral

Since \(g(x)=g_1\) on the interval \((a,d)\), we have: $$ \int_a^d f(x)dg(x) = \int_a^d f(x) dg_1 $$ As \(g_1\) is a constant, the integral simplifies to: $$ \int_a^d f(x) dg_1 = g_1 [f(d-) - f(a+)] $$ where \(f(d-)\) is the limit of \(f(x)\) as \(x\) approaches \(d\) from the left, and \(f(a+)\) is the limit of \(f(x)\) as \(x\) approaches \(a\) from the right.

Computing the second integral

Similarly, since \(g(x)=g_2\) on the interval \((d, b)\), we have: $$ \int_d^b f(x) dg(x) = \int_d^b f(x) dg_2 $$ Again, as \(g_2\) is a constant, the integral simplifies to: $$ \int_d^b f(x) dg_2 = g_2 [f(b-) - f(d+)] $$ where \(f(b-)\) is the limit of \(f(x)\) as \(x\) approaches \(b\) from the left, and \(f(d+)\) is the limit of \(f(x)\) as \(x\) approaches \(d\) from the right.

Adding the integrals

Now that we have computed both integrals, we can add them together to get the value of the integral \(\int_{a}^{b} f(x) d g(x)\): $$ \int_{a}^{b} f(x) d g(x)= g_1 [f(d-) - f(a+)] + g_2 [f(b-) - f(d+)] $$ Thus, the value of the integral exists, and we have found its value.

Key concepts

Piecewise Constant Functions

When we talk about piecewise constant functions, we're dealing with functions that are constant over specific intervals. In the given problem, the function \( g(x) \) is defined as piecewise constant because:

• For \( a < x < d \), the function takes the constant value \( g_1 \).
• For \( d < x < b \), it jumps to another constant value \( g_2 \).

This type of function is helpful because it simplifies calculations in integrals due to the constant nature over segments. It only changes values at discrete points, such as \( a \), \( d \), and \( b \) here, making it straightforward to handle when setting up integrals like the Riemann-Stieltjes integral. Each segment is individually simple to integrate because the function doesn't change within those bounds.

Continuity

Continuity is a critical property of functions addressed in this exercise. A function is continuous if, roughly speaking, there are no sudden jumps or breaks in its values. Here’s how continuity applies to the function \( f(x) \):

• It's continuous from the right at \( a \), which means for values approaching \( a \) from just above, the function doesn't jump suddenly.
• It's continuous from the left at \( b \), ensuring no abrupt changes as we approach \( b \) from below.
• Lastly, the function is continuous at \( d \), meaning it doesn't jump when crossing \( d \).

These continuity conditions make it manageable to compute integrals, like in the Riemann-Stieltjes approach, because they ensure the function behaves nicely over the specified interval, allowing for straightforward limits and integral calculations.

Riemann-Stieltjes Integral

The Riemann-Stieltjes integral generalizes the traditional Riemann integral by integrating one function, \( f(x) \), with respect to another, \( g(x) \). In this context, the integral \[\int_{a}^{b} f(x) d g(x)\]is significant because it incorporates how both functions interact over the interval \([a, b]\). Instead of measuring area just under \( f(x) \), it considers how \( g(x) \) changes.In the exercise, \( g(x) \) alters stepwise at \( d \), marking breaks at specific points, while \( f(x) \)'s continuity ensures no abrupt values around these changes. This integral breaks into two parts:

• \( \int_a^d f(x) d g(x) \) where \( g(x) \) is constant and simplifies due to this constant nature.
• \( \int_d^b f(x) d g(x) \) following a similar simplification.

Ultimately, the Riemann-Stieltjes integral exists due to this structured setup, and combining results from sub-intervals gives us the complete value, clearly connecting how piecewise behavior of \( g(x) \) aligns with the smooth continuity of \( f(x) \).