Problem 2 - A (terse) introduction to Lebesgue integration

Question

Prove that the absolute value of a Riemann integrable function is Riemann integrable.

Step-by-Step Solution

Answer

The absolute value of a Riemann integrable function is Riemann integrable.

Step 1: Define Riemann Integrable Function

A function \( f \) is said to be Riemann integrable on an interval \([a, b]\) if for every \( \epsilon > 0 \), there exist partitions \( P \) of \( [a, b] \) such that the upper sum \( U(f, P) \) and the lower sum \( L(f, P) \) satisfy \( U(f, P) - L(f, P) < \epsilon \).

Step 2: Understand Absolute Value Property

The absolute value of a function, \( |f| \), which is defined by \( |f(x)| = \max(f(x), -f(x)) \), retains the characteristic of the original function, but it is always non-negative.

Step 3: Prove \(|f|\) is Bounded on Partition

Since \( f \) is Riemann integrable, \( f \) is bounded on \([a, b]\). Therefore, \(|f|\) is also bounded, as \( |f(x)| \leq M \) for some \( M > 0 \) and for all \( x \in [a, b] \).

Step 4: Consider Partitions for Riemann Integrability of \(|f|\)

Let \( \epsilon > 0 \) be given. Since \( f \) is Riemann integrable, there exists a partition \( P \) such that \( U(f, P) - L(f, P) < \epsilon \). The same partition can be used to analyze \( |f| \).

Step 5: Relate \( U(|f|, P) \) and \( L(|f|, P) \)

Since \( |f(x)| \leq M \) and \( |f(x)| \geq 0 \), calculate the upper and lower sums for \(|f|\) using the same partition \( P \) and note that \( U(|f|, P) - L(|f|, P) \leq U(f, P) - L(f, P) < \epsilon \).

Step 6: Conclude \(|f|\) is Riemann Integrable

Given \( U(|f|, P) - L(|f|, P) < \epsilon \), we conclude that \(|f|\) is also Riemann integrable on \([a, b]\) since we can make the difference between the upper and lower sums for \(|f|\) as small as desired.

Key Concepts

Absolute Value Function

The absolute value function, denoted as \(|x|\), transforms any real number to its non-negative counterpart. For a function \(f(x)\), the absolute value is expressed as \(|f(x)| = \max(f(x), -f(x))\). This ensures that the result is always zero or greater, regardless of the input value. This property makes the absolute value function very useful in calculus, particularly when working with Riemann integrals. The essence here is maintaining the nature of \(f(x)\) while ensuring all outputs are non-negative, thus simplifying the analysis when determining integrability.

Upper Sum and Lower Sum

To determine if a function is Riemann integrable, we use the concepts of upper and lower sums. These sums help measure the area under a curve, which is crucial for integration.

• The **upper sum** \(U(f, P)\) calculates an overestimate of this area. It's the sum of areas of rectangles that sit above the curve.
• The **lower sum** \(L(f, P)\) provides an underestimate. It's the sum of areas of rectangles under the curve.

If for any \(\epsilon > 0\), there exists a partition \(P\) such that \(U(f, P) - L(f, P) < \epsilon\), the function is integrable. The closeness of these sums indicates how well the rectangles approximate the area below \(f(x)\).

Bounded Functions

A function is bounded on an interval if it can be confined within some limits, not going to infinity. For Riemann integrable functions, being bounded is a key property. It means there are numbers \(m\) and \(M\) such that \(m \leq f(x) \leq M\) for all \(x\) in a certain interval.
This boundedness criterion is essential in proving integrability, ensuring that the function does not have any unmanageable spikes or dips that could make integration impossible. When discussing the absolute value, the bounded nature of \(f\) guarantees that \(|f|\) is also confined within a specific range, hence itself being bounded.

Partitions in Integration

Partitions divide the interval into smaller sub-intervals, essential for calculating upper and lower sums.
A partition \(P\) of an interval \([a, b]\) is a finite sequence of numbers \(a = x_0 < x_1 < ... < x_n = b\). These partitions help refine the area approximation under a curve.
The choice of partition affects how precisely the upper and lower sums approximate the integral of the function.
By choosing sufficiently small sub-intervals in \(P\), the difference between \(U(f, P)\) and \(L(f, P)\) can be minimized to be less than any allocated \(\epsilon\), demonstrating the function's integrability. Thus, partitions are a crucial tool in the process of showing a function is Riemann integrable.