I have been trying to make sense of the existence of two integration theories, that of Riemann and Lebesgue. The latter integrates the Dirichlet function and assigns a value 1, while the former theory cannot integrate the function. What is integration and what does the value of integral mean? Should it be understood as the existence of a linear functional with good limit-properties on as large a class of functions as possible? Given this discrepancy between Riemann's theory and Lebesgue's theory, how does it turn out that by applying both the theories one gets, for instance, that the sum of reciprocals of squares of integers is equal to $\pi^2/6$?
The notion of 'powerful' theory in algebra is related to the ability to solve more and more complicated questions, by discovering new notions and their relationships. But analysis seems to me to include 'choice' between two theories to solve a problem, in which case what does it mean to solve a problem in analysis?
I am sorry if I am not able to put forth my questions clearly.
