The Lebesgue Integral

Marcus Littleton              

When discussing integrals in calculus and mathematics courses of a similar level, it is commonly assumed that the integrals in question are Riemann integrable. Many mathematicians will spend the majority of their studies assuming the Riemann integral is the only applicable kind of integral but there is another that is scarcely considered, the Lebesgue integral. The Lebesgue integral is initially disregarded in introductory understandings of integrals because Lebesgue integration also requires an understanding of measure and measure theory. This use of measure when evaluating a function allows the Lebesgue integral to calculate a wider array of functions than the Riemann integral can and allows for broader domains on which said functions can be defined. This is most apparent for the Dirichlet function, a function defined to be 0 when it’s argument is irrational and 1 otherwise. This is due to the Dirichlet function not being Riemann integrable by the nature of irrational numbers but, under the consideration of Lebesgue measure, the function is Lebesgue integrable. My capstone project aims to give an explanation on how to obtain the Lebesgue integral by integrating with respect to measure while comparing it to and differentiating it from the Riemann integral.

• Marcus Littleton is obtaining a Bachelor of Science in Mathematics with the goal to understand and gain the necessary skills to apply higher levels of mathematics to computer programming. After graduation, he aims to pursue opportunities in Data Science, Data Analysis, or in Software Development where he can apply his background in mathematics in meaningful ways.