Problem 41 - A Course in Calculus and Real Analysis

Question

Let \(f:[a, b] \rightarrow \mathbb{R}\) be a bounded function. For \(c \in(a, b)\), let \(f_{1}\) and \(f_{2}\) denote the restrictions of \(f\) to the subintervals \([a, c]\) and \([c, b]\) respectively. Prove the following: (i) \(L(f)=L\left(f_{1}\right)+L\left(f_{2}\right)\), (ii) \(U(f)=U\left(f_{1}\right)+U\left(f_{2}\right)\). [Note: The results in (i) and (ii) are refined versions of Proposition \(6.7\), and may be referred to as Domain Additivity of Lower Riemann Integrals and Domain Additivity of Upper Riemann Integrals, respectively.]

Step-by-Step Solution

Answer

To prove the domain additivity of lower and upper Riemann integrals, we first recall their definitions and represent them as \(L(f) = \sup_{P\in \mathcal{P}} L_P(f)\) and \(U(f) = \inf_{P\in \mathcal{P}} U_P(f)\). For \(f_1\) on \([a, c]\) and \(f_2\) on \([c, b]\), we calculate their lower/upper sums and connect them to that of \(f\). By considering arbitrary partitions and establishing inequalities relating \(L(f)\), \(L(f_1)\), and \(L(f_2)\), we prove the domain additivity of the lower integrals, \(L(f) = L(f_1) + L(f_2)\). Following the same steps and using the upper sums instead of the lower sums, we prove the domain additivity of the upper integrals, \(U(f) = U(f_1) + U(f_2)\).

Step 1: Recall the definition of the lower integral

First, we recall that the lower integral of a bounded function \(f\) on an interval \([a, b]\) is defined as: \[L(f) = \sup_{P\in \mathcal{P}} L_P(f),\] where \(\mathcal{P}\) is the set of all partitions of \([a, b]\) and \(L_P(f)\) is the lower sum associated with each partition \(P\).

Step 2: Identify lower sums for the subintervals

We must now identify the lower sums for \(f_1\) on \([a, c]\) and \(f_2\) on \([c, b]\): \[L_1(P_1) = \sum_{i=1}^n m_i(t_i - t_{i-1}),\] and \[L_2(P_2) = \sum_{i=1}^m M_i(s_{i} - s_{i-1}),\] where \(m_i\) and \(M_i\) are the infimum of \(f_1\) and \(f_2\) on each subinterval, and \(P_1\) and \(P_2\) are partitions of \([a, c]\) and \([c, b]\), respectively.

Step 3: Connect the lower integrals of the subintervals

Now, let \(P\) be a common partition for both \([a, c]\) and \([c, b]\), such that \(P_1\) and \(P_2\) are refinements of the partition \(P\). The lower sum associated with \(P\) for \(f\) is given by: \[L_P(f) = \sum_{i=1}^n m_i(t_i - t_{i-1}) +\sum_{i=1}^m M_i(s_{i} - s_{i-1}).\] Since the partition \(P\) is arbitrary, we can write: \[L(f) \geq L_1(P_1) + L_2(P_2)\] for every pair of partitions \(P_1\) and \(P_2\). Now, we have: \[L(f) \geq \sup_{P_1 \in \mathcal{P}_1} L_1(P_1) + \sup_{P_2 \in \mathcal{P}_2} L_2(P_2) = L(f_1) + L(f_2).\]

Step 4: Connect the lower integrals of the function and its restrictions

We also have the following inequalities due to the fact that \(f_1\) and \(f_2\) are restrictions of \(f\): \[L(f_1) \leq L(f)\] and \[L(f_2) \leq L(f).\] By adding these inequalities, we get: \[L(f_1) + L(f_2) \leq L(f).\]

Step 5: Combine the inequalities to prove domain additivity

From steps 3 and 4, we have the following inequalities: \[L(f) \geq L(f_1) + L(f_2)\] and \[L(f) \leq L(f_1) + L(f_2).\] These two inequalities imply the domain additivity of the lower integrals: \[L(f) = L(f_1) + L(f_2).\] (ii) Proving the domain additivity of upper Riemann integrals:

Step 6: Recall the definition of the upper integral

First, we recall that the upper integral of a bounded function \(f\) on an interval \([a, b]\) is defined as: \[U(f) = \inf_{P\in \mathcal{P}} U_P(f),\] where \(\mathcal{P}\) is the set of all partitions of \([a, b]\) and \(U_P(f)\) is the upper sum associated with each partition \(P\).

Step 7: Follow a similar process as for the lower integrals

Proving the domain additivity of the upper Riemann integrals follows a very similar process as we used for the lower integrals. We will consider the upper sum associated with \(P\) for \(f\), \(U_P(f)\), and prove that: \[U(f) = U(f_1) + U(f_2).\] We will again make use of the relationships between the function \(f\) and its restrictions \(f_1\) and \(f_2\) on the subintervals. By following the same reasoning as in steps 1 through 5 above, using the upper sums instead of the lower sums, we can establish the domain additivity of the upper Riemann integrals.

Key Concepts

Lower Riemann Integral

Understanding the Lower Riemann Integral involves visualizing how to approximate the area under a curve for a given bounded function. Imagine a function \( f(x) \) defined over the interval \([a, b]\). We want to calculate the area under this curve using what we call *lower sums*. These are sums that underestimate the area.

To find the lower sum, divide the interval \([a, b]\) into smaller subintervals. For each subinterval, determine the infimum (the greatest lower bound) of \( f(x) \). Multiply each infimum by the width of its corresponding subinterval. Adding up these products gives the lower sum, \( L_P(f) \), for partition \( P \).

The lower Riemann integral, \( L(f) \), is then the supremum (the least upper bound) of these lower sums, considering all possible partitions of \([a, b]\). The goal is to find the maximum possible lower sum that still fits "under" the curve of \( f(x) \). This provides one way to approximate the integral of \( f \) over \([a, b]\). Remember:

• The lower sum always underestimates the area.
• More partitions lead to a better approximation.
• Supremum of lower sums gives the lower Riemann integral.

Upper Riemann Integral

The Upper Riemann Integral works similarly to the lower integral but instead aims to overestimate the area under the curve of a bounded function \( f \). It offers another method of approximation by using *upper sums*.

Again, consider dividing the interval \([a, b]\) into subintervals. For each subinterval, identify the supremum (the least upper bound) of \( f(x) \). Multiply this supremum by the width of its subinterval. Summing these products yields the upper sum, \( U_P(f) \), for partition \( P \).

The upper Riemann integral, \( U(f) \), then is the infimum (greatest lower bound) of all these upper sums across every possible partition of \([a, b]\). Here's what you need to keep in mind:

• Upper sums always overestimate the area.
• As you add more subintervals, your approximation improves.
• The infimum of these upper sums gives you the upper Riemann integral.

This mechanism of estimating from above ensures that alongside the lower integral, the closer the two are, the more precise your approximation of the integral becomes.

Domain Additivity

Domain Additivity is an essential principle in Riemann Integration for handling intervals. It states that the integral over an entire interval \([a, b]\) can be broken into smaller subintervals, and the sum of these integrals across subintervals will equal the integral over the entire interval.

For a bounded function \( f(x) \), if you have two subintervals \([a, c]\) and \([c, b]\), domain additivity tells us:

• The lower integral over \([a, b]\) is the sum of the lower integrals over \([a, c]\) and \([c, b]\).
• Similarly, the upper integral over \([a, b]\) is the sum of the upper integrals over \([a, c]\) and \([c, b]\).

Breaking down the interval in this manner enables calculations to be more manageable and is especially useful when dealing with more complex functions across multiple segments. This property ensures continuity and consistency with the whole function's integral, emphasizing:

• The integrability of sections relies on the entire interval's integrability.
• Being able to employ this additivity can simplify problem-solving in integration.

This principle is crucial for validating consistent and accurate calculations when splitting areas under curves into distinct, manageable pieces.