Exploring the Integral of Bounded Functions: Lebesgue vs Riemann

Understanding the integration of bounded functions over an interval can be intricate. This article delves into the differences and nuances between Riemann and Lebesgue integrals, providing clarity on which types of bounded functions exhibit integrability issues.

1. Integrability of Bounded Functions: Riemann and Lebesgue

In the realm of real analysis, the question of whether the integral of a bounded function over an interval is always defined is not straightforward. It involves delving into the intricacies of Riemann and Lebesgue integrals, two of the most widely used methods in mathematical analysis.

1.1 Riemann Integrability

Riemann integration is arguably the more intuitive and foundational approach. By the late 19th century, it was established that continuity was a key factor for Riemann integrability. Specifically, a bounded function f:[a, b] to mathbb{R} is Riemann integrable if and only if the set of discontinuity points is of measure zero. This means that the function f is integrable if the set of its discontinuities is negligible.

The set of points of continuity of f is denoted by C_f. Thus, f is Riemann integrable if and only if [a, b] setminus C_f is a set of Lebesgue measure zero. If there are any subintervals [c, d] subseteq [a, b] where f is everywhere discontinuous, then f cannot be Riemann integrable.

1.2 Lebesgue Integrability

Lebesgue integration, introduced to address the limitations of Riemann integration, is more inclusive. It was designed to integrate a broader class of functions, including those that are nowhere continuous. Lebesgue integrability is less restrictive; even discontinuous functions can be integrated under certain conditions.

A bounded function f:[a, b] to mathbb{R} is Lebesgue integrable if the set of its discontinuities is of measure zero in the sense of Lebesgue measure. This means that the function f can be integrated without the continuity requirements imposed by Riemann integration.

1.3 Differentiability and Integrability

For a differentiable function F:[a, b] to mathbb{R} with a bounded derivative f F', the Riemann integral would yield int_a^b f(x) dx F(b) - F(a). However, in the 1880s, it was discovered that such a function may not be Riemann integrable if it has too many discontinuities. Lebesgue integration, however, handles this scenario effectively.

2. Constructing Non-Integrable Functions

To construct a bounded function that is not Riemann integrable, ensure that there are no dense sets of points of continuity. A counterexample can be created by making the set of discontinuity points dense in the interval [a, b]. This is a sophisticated construction, often leveraging the Axiom of Choice, which allows for the creation of non-measurable sets.

By the end of the 19th century, it was known that a bounded function f:[a, b] to mathbb{R} is Riemann integrable if and only if [a, b] setminus C_f is a set of Lebesgue measure zero. This means that the function must have a negligible set of discontinuities for Riemann integrability to hold.

3. Set Theoretic Real Analysis and the Axiom of Choice

Set-theoretic real analysis uses the tools of modern set theory to study real functions and is particularly interested in pathological objects. The Axiom of Choice can be used to construct bounded functions that are not Lebesgue integrable. If you believe in the Axiom of Choice, it is possible to construct a bounded function that is not Lebesgue integrable, as Solovay demonstrated in 1970.

However, if you do not believe in the Axiom of Choice, it still does not guarantee that all bounded functions are Lebesgue integrable. Solovay showed that it is possible to construct a model of set theory where every set of reals is Lebesgue measurable, but this is a highly specialized and complex topic in set theory.

Conclusion

The integration of bounded functions over an interval is a complex topic that involves understanding the nuances of Riemann and Lebesgue integrals. Riemann integration relies on the continuity of a function, while Lebesgue integration is more inclusive and can handle a broader range of functions, including those that are discontinuous. The Axiom of Choice plays a significant role in these constructions, highlighting the importance of foundational mathematical assumptions in more advanced topics.