Question

For the case where \(g\) is nondecreasing and \(f\) is bounded on \([a, b],\) define upper and lower Riemann-Stieltjes integrals in a way analogous to Definition 3.1 .3 .

Short answer

Question: Define the upper and lower Riemann-Stieltjes integrals for functions \(f\) and \(g\) over an interval \([a, b]\), given that \(g\) is nondecreasing and \(f\) is bounded on this interval. Answer: The upper Riemann-Stieltjes integral, \(\overline{\int_a^b} f d g\), is defined as the infimum of the set of all upper Riemann-Stieltjes sums (\(U(f, g, P)\)) for all possible partitions \(P\) of the interval \([a, b]\). The lower Riemann-Stieltjes integral, \(\underline{\int_a^b} f d g\), is defined as the supremum of the set of all lower Riemann-Stieltjes sums (\(L(f, g, P)\)) for all possible partitions \(P\) of the interval \([a, b]\).

Step-by-step solution

Understanding Riemann-Stieltjes integral

Riemann-Stieltjes integral is an extension of the Riemann integral, where the function \(g\) can be any nondecreasing function on the interval \([a, b]\), instead of only uniform partitions as in the Riemann integral. We will define the upper and lower Riemann-Stieltjes integrals for the functions \(f\) and \(g\) on the interval \([a, b]\).

Define a Partition of the Interval

First, we'll start by defining a partition of the interval \([a, b]\). A partition \(P\) consists of a finite set of points {\(x_0, x_1, x_2, ..., x_n\)} such that \(a = x_0 < x_1 < x_2 < ... < x_{n-1} < x_n = b\). Here, \(n\) is the number of sub-intervals created by the partition.

Define the Upper and Lower Riemann-Stieltjes Sums

For each sub-interval [\(x_{i-1}, x_i\)], let \(M_i = sup\){\(f(x) : x_{i-1} \le x \le x_i\)} and \(m_i = inf\){\(f(x) : x_{i-1} \le x \le x_i\)}. These represent the greatest and smallest values of the function \(f\) on each sub-interval, respectively. Now, we define the upper and lower Riemann-Stieltjes sums as follows: $$U(f, g, P) = \sum_{i=1}^{n} M_i [g(x_i) - g(x_{i-1})]$$ $$L(f, g, P) = \sum_{i=1}^{n} m_i [g(x_i) - g(x_{i-1})]$$ Here, \(U(f, g, P)\) and \(L(f, g, P)\) represent the upper and lower Riemann-Stieltjes sums, respectively.

Define the Upper and Lower Riemann-Stieltjes Integrals

The upper Riemann-Stieltjes integral, denoted by \(\overline{\int_a^b} f d g\), is defined as the infimum of the set of all upper Riemann-Stieltjes sums (\(U(f, g, P)\)) for all possible partitions \(P\) of the interval \([a, b]\). In other words, $$\overline{\int_a^b} f d g = inf\({\)U(f, g, P) : P\( is any partition of \)[a, b]$}. Similarly, the lower Riemann-Stieltjes integral, denoted by \(\underline{\int_a^b} f d g\), is defined as the supremum of the set of all lower Riemann-Stieltjes sums (\(L(f, g, P)\)) for all possible partitions \(P\) of the interval \([a, b]\). In other words, $$\underline{\int_a^b} f d g = sup\({\)L(f, g, P) : P\( is any partition of \)[a, b]$}.

Riemann-Stieltjes Integrability Condition

A function \(f\) is said to be Riemann-Stieltjes integrable with respect to \(g\) on the interval \([a, b]\) if and only if the upper and lower Riemann-Stieltjes integrals are equal, i.e., $$\overline{\int_a^b} f d g = \underline{\int_a^b} f d g$$ In this case, the common value of the upper and lower Riemann-Stieltjes integrals is called the Riemann-Stieltjes integral of \(f\) with respect to \(g\) on the interval \([a, b]\), denoted by \(\int_a^b f d g\).

Key concepts

Upper and Lower Riemann-Stieltjes Integrals

Imagine slicing a sub sandwich into equally spaced pieces, where each slice varies in thickness according to a specific rule. This visual can help us understand the Upper and lower Riemann-Stieltjes integrals. These integrals are a generalization of the Riemann integrals, where instead of looking at the area under a curve, we measure how a function f accumulates based on another function g.

Given a bounded function f and a nondecreasing function g on an interval <[a, b]>, we start by calculating the supremum (largest value) and infimum (smallest value) of f in subintervals created by a partition of the interval. Then, for the upper sum, we multiply those maximum values by the increase of g at each subinterval, and for the lower sum, we do the same with the minimum values. The sum of all products in both scenarios gives us the upper and the lower Riemann-Stieltjes sums. As we refine the partition to make slices thinner, we approach the upper and lower Riemann-Stieltjes integrals, which are the infimum and supremum of all possible upper and lower sums, respectively.

Partition of an Interval

A partition of an interval is akin to a detailed itinerary for a road trip, breaking down a long journey into shorter legs. Specifically, for an interval [a, b], a partition is a sequence of checkpoints, or partition points, that split the interval into smaller contiguous subintervals.

Each partition brings with it a unique set of these points
{x_0, x_1, ..., x_n}, where a = x_0 < x_1 < ... < x_n = b. Think of these as stops along the route from point a to point b. The more checkpoints there are, the finer the partition, allowing for a more detailed analysis of how function f behaves as influenced by function g across the interval. When we talk about taking all possible partitions, we essentially consider every conceivable way to break up the interval, akin to plotting every possible stop between two destinations.

Riemann-Stieltjes Integrability Condition

To determine if we can calculate the Riemann-Stieltjes integral for a function f with respect to another function g, there is an essential checkpoint—the Riemann-Stieltjes Integrability Condition. This condition is like a 'green light' that signals when the journey of calculation is feasible.

For a function f to be Riemann-Stieltjes integrable relative to g over an interval <[a, b]>, there must be agreement between the overall greatest and least possible accumulations (the upper and lower Riemann-Stieltjes integrals, respectively). When these two integrals match, we've hit the jackpot: f is integrable with respect to g, and the common value is what we call the Riemann-Stieltjes integral. This convergence is a crucial litmus test for integrability and ensures that the function f, observed through the lens of g's variations, behaves consistently enough to allow for meaningful accumulation measurement.

Supremum and Infimum of Functions

In mathematics, just as in life, we often look for extremes—the highest and the lowest points. The concepts of supremum and infimum are the mathematical embodiment of this search when dealing with functions. The supremum is essentially the 'ceiling' of function values within a subinterval, whereas the infimum is the 'floor.'

To find the supremum of a function f over some interval, we scout for the highest value that f ever reaches, without crossing it—even if f doesn't actually attain this value itself. Conversely, the infimum is the greatest value that is never exceeded by the function within the interval.

These concepts are central in defining the upper and lower Riemann-Stieltjes sums because they represent the bounds of f's behavior within each slice of the partition. Think of them as guidelines that contain the function's values, highlighting its extremes as we seek to measure how it integrates with respect to another function across an interval.