Problem 34 - A Course in Calculus and Real Analysis

Question

Let \(f:[a, b] \rightarrow \mathbb{R}\) be an integrable function. If \(\left(P_{n}\right)\) is a sequence of partitions of \([a, b]\) such that \(\mu\left(P_{n}\right) \rightarrow 0\), then show that \(U\left(P_{n}, f\right)-L\left(P_{n}, f\right) \rightarrow\) 0. Is the converse true?

Step-by-Step Solution

Answer

For the given function \(f\), which is integrable on \([a, b]\), and a sequence of partitions \(\left(P_{n}\right)\) with \(\mu\left(P_{n}\right) \rightarrow 0\), we have shown that \(U\left(P_{n}, f\right) - L\left(P_{n}, f\right) \rightarrow 0\). However, the converse is not true. The function \(f(x) = 1\) on the interval \([0, 1]\) serves as a counterexample, where the difference between upper and lower sums is always 0, but the mesh size \(\mu(P_n)\) can remain constant and nonzero.

Step 1: Upper and lower sums definition

Given a partition \(P\) of \([a, b]\) and an integrable function \(f\), the upper sum, denoted as \(U(P, f)\), is the sum of the product of the widths of the partition's subintervals and the supremum of the function in each subinterval. Similarly, the lower sum, denoted by \(L(P, f)\), is the sum of the product of the widths of the partition's subintervals and the infimum of the function in each subinterval. Formally, if \(P: a = x_0 < x_1 < \dots < x_n = b\), then: \[U(P, f) = \sum_{i=1}^n M_i\Delta x_i\] \[L(P, f) = \sum_{i=1}^n m_i\Delta x_i\] where \(\Delta x_i = x_i - x_{i-1}\), \(M_i = \sup\{f(x): x_{i-1}\leq x \leq x_i\}\) and \(m_i = \inf\{f(x): x_{i-1}\leq x \leq x_i\}\).

Step 2: Relation between upper and lower sums

Since the function \(f\) is integrable, we know that for any \(\epsilon > 0\), there exists a partition \(P_\epsilon\) such that \(U(P_\epsilon, f) - L(P_\epsilon, f) < \epsilon\). This relation is important to express upper and lower sums with respect to the partitions of the given sequence.

Step 3: Show that the difference between upper and lower sums tends to 0

Now we have a sequence of partitions \(\left(P_{n}\right)\), and we are given that \(\mu\left(P_{n}\right) \rightarrow 0\). We need to show that \(U\left(P_{n}, f\right) - L\left(P_{n}, f\right) \rightarrow 0\). Since \(f\) is integrable, given any \(\epsilon > 0\), there exists a partition \(P_\epsilon\) such that \(U(P_\epsilon, f) - L(P_\epsilon, f) < \epsilon\). Now, choose some \(N\) in the sequence such that when \(n \geq N\), \(\mu(P_n) \leq \mu(P_\epsilon)\). Then, we have that for any \(n \geq N\), \(U(P_n, f) - L(P_n, f) \leq U(P_\epsilon, f) - L(P_\epsilon, f) < \epsilon\). As a result, since we can find such an \(N\) for any \(\epsilon > 0\), we have established that \(U\left(P_{n}, f\right) - L\left(P_{n}, f\right) \rightarrow 0\).

Step 4: Discuss the converse

Now let's discuss whether the converse of the statement is true, i.e., if \(U\left(P_{n}, f\right) - L\left(P_{n}, f\right) \rightarrow 0\), then is it true that \(\mu\left(P_{n}\right) \rightarrow 0\)? The answer is no. The counterexample can be the function \(f(x) = 1\) on the interval \([0, 1]\). For any partition \(P_n\) of \([0, 1]\), we have \(M_i = m_i = 1\) for all \(i\). Hence, \(U(P_n, f) = L(P_n, f) = 1\). So \(U(P_n, f) - L(P_n, f) = 0\) for any \(n\), but \(\mu(P_n)\) can remain constant and nonzero. Thus, the converse is not true.

Key Concepts

Upper Sum

When dealing with Riemann Integration, the concept of the Upper Sum is a fundamental aspect. It's a method used to approximate the area under a curve, which represents an integral. The Upper Sum, denoted by \( U(P, f) \), is calculated by taking a partition \( P \) of a given interval \([a, b]\) and looking at each subinterval. For each subinterval, you find the supremum (the least upper bound) of the function within that range. This supremum is then multiplied by the width of that subinterval. By summing up the products for all subintervals, you obtain the Upper Sum of the function over that partition.

• For each subinterval, the supremum represents the highest point of the function within that subinterval.
• Thus, the Upper Sum often overestimates the true area under the function, especially where the function has peaks.
• This overestimation is why it is called "Upper".

Understanding this concept allows you to comprehend how integrals can be approximated from above.

Lower Sum

In contrast to the Upper Sum, the Lower Sum offers a different approach for approximating integrals. It is useful in estimating from below the area under the curve defined by a function. The Lower Sum, \( L(P, f) \), involves finding the infimum (the greatest lower bound) of the function over each subinterval of a given partition \( P \) of the interval \([a, b]\). Each infimum is multiplied by the width of its respective subinterval, and the sum of these products gives the Lower Sum.

• Each infimum is the smallest value the function reaches within a subinterval.
• Typically, the Lower Sum underestimates the area because it takes the lowest point per subinterval.
• The word "Lower" signifies this tendency to underestimate.

Recognizing the role of Lower Sums is vital for understanding the concept of finding bounds for integrals.

Partition Limit

A critical component of ensuring accurate approximations in Riemann Integration is understanding the Partition Limit. This concept pertains to the idea of partitioning an interval with finer and finer subdivisions. As you refine the partitions of the interval \([a, b]\) by reducing the maximum width of subintervals, denoted as \( \mu(P_n) \), you approach the true value of an integral.The Partition Limit is significant because:

• As the partitions become infinitely small, both the Upper and Lower Sums converge towards the true value of the Riemann Integral.
• For a function to be integrable, the difference between the Upper and Lower Sums must tend to zero as the partition is refined (i.e., \( \mu(P_n) \rightarrow 0 \)).
• This condition ensures that the integral of the function exists and can be accurately computed.

Understanding how the Partition Limit works helps in visualizing the process of integration in practical terms.

Integrable Function

The concept of an Integrable Function is central to calculus and Riemann Integration. An integrable function is one whose integral over an interval can be determined accurately. This is possible when the difference between the Upper Sum \( U(P_n, f) \) and Lower Sum \( L(P_n, f) \) approaches zero as you refine the partition (by letting \( \mu(P_n) \rightarrow 0 \)).
For a function \( f \) on a closed interval \([a, b]\), being integrable means:

• The area under the curve \( f \) is well-defined and can be approximated using partitions.
• The function doesn't have too many wild oscillations or discontinuities over \([a, b]\) that might make the integration undefined.
• It is possible to sandwich the actual integral value between the Upper and Lower Sums.

This concept is integral (pun intended!) to determining when and how we can compute integrals precisely.