This book presents a not-quite-standard approach to measure theory and the theory of the Lebesgue integral. Historical questions that lead to the development of these theories are the motivation for preliminary considerations and form into a starting point of the presentation. In particular, problems of whether the Fourier expansion of a function converges to the function, the relationship between integration and differentiation, and the relationship between continuity and differentiation are discussed at the very beginning of the book. The Riemann integral is presented as the first concept of integration, not only to show its construction but also to show how broadly one could define a function that is still integrable. The story then continues with a description of certain properties of the real axis and problems with the Fundamental Theorem of Calculus and term-by-term integration, finishing with measure theory and the Lebesgue integral, and Fourier series. The way that facts are presented makes the book accessible for graduate or advanced undergraduate students as an alternative to the standard approach of teaching real analysis. The book will be interesting for teachers as well.