Proving the Riemann Integrability of the Product of Two Riemann Integrable Functions

Introduction

When dealing with Riemann integrable functions, one important question is whether the product of two such functions is also Riemann integrable. This problem is crucial in both theoretical and applied mathematics. According to Lebesgue's theorem on Riemann integration, under certain conditions, the product of two Riemann integrable functions is also Riemann integrable. In this article, we will delve into the proof of this statement and explore the conditions under which the product function is Riemann integrable.

Background and Definitions

First, let's define some terms and background information. A function f is said to be Riemann integrable on the interval [a, b] if the upper and lower Riemann sums converge to the same value as the partition of the interval becomes finer. This function must also be bounded and have a set of discontinuities of measure zero. Lebesgue's theorem on Riemann integration states that if two functions f and g are Riemann integrable on [a, b], then their product fg is also Riemann integrable on the same interval under certain conditions.

Understanding Lebesgue's Theorem on Riemann Integration

Lebesgue's theorem provides a powerful tool for proving the Riemann integrability of the product of two functions. According to this theorem, if f is Riemann integrable on [a, b], it can be shown that the set of points where f is discontinuous has measure zero. Similarly, if g is Riemann integrable on [a, b], the set of points where g is discontinuous also has measure zero.

Key Definitions and Theorems

1. Measure Zero: A set of points has measure zero if, for any positive number ε, it can be covered by a countable number of intervals whose total length is less than ε.

2. Continuity and Discontinuity: A function f is continuous at a point if the limit of f(x) as x approaches that point equals the value of f at that point.

Proof of Riemann Integrability of the Product

Now, let's prove that if f and g are Riemann integrable on [a, b], then their product fg is also Riemann integrable on the same interval.

Step 1: Identifying the Sets of Discontinuity

Consider the sets A and B of measure zero, where A and B are the sets of points where f and g are discontinuous, respectively. According to Lebesgue's theorem, these sets have measure zero. Therefore, the union of these sets, A ∪ B, also has measure zero.

Step 2: Continuity of the Product Function

When x is outside the set A ∪ B, both f(x) and g(x) are continuous. Since the product of two continuous functions is also continuous, fg(x) is continuous for all x outside the set A ∪ B.

Step 3: Riemann Integrability of the Product Function

Given that the set A ∪ B has measure zero, and fg is continuous outside this set, it follows that fg is Riemann integrable on the interval [a, b]. This is because the discontinuities of fg form a set of measure zero, which is the required condition for Riemann integrability.

Examples and Applications

Let's consider an example to illustrate the application of this theorem. Suppose we have two functions f(x) sin(x) and g(x) cos(x) on the interval [0, π]. Both functions are Riemann integrable on [0, π] because they are continuous on this interval. The product of these functions, fg(x) sin(x)cos(x) (1/2)sin(2x), is also continuous on [0, π] and hence Riemann integrable on this interval.

Trivial Boundaries and Discontinuity Sets

Furthermore, the functions f(x) and g(x) are trivially bounded and their sets of discontinuities are of zero measure because they are part of the union of the sets where fg is discontinuous. This is consistent with Lebesgue's theorem, which states that the product of two Riemann integrable functions is Riemann integrable if the set of their discontinuities has measure zero.

Conclusion

In conclusion, Lebesgue's theorem on Riemann integration provides a powerful framework for proving the Riemann integrability of the product of two Riemann integrable functions. By identifying the sets of discontinuity and using the properties of set measure, one can establish the Riemann integrability of the product function. This result is significant in various areas of mathematics, including analysis and applied mathematics, where the integrability of composite functions is often required.