This is a question about Lebesgue's theory of integration. What is the motive behind giving a meaning to the integral to as larger a class of functions as possible? What is the motive behind looking for a "good" integration theory? So far I am accustomed to mathematical theories that involve constructing new subsets and finding patterns among those subsets in solving some concrete questions. But, analysis seems to be a little different in that it is primarily based on giving meaning to operations on infinitely many quantities. In this context, I find it hard to understand how Lebesgue might have thought about coming up with a new theory when Riemman's theory was already in place. What exactly is the fault of the analysis that it can't integrate Dirichlet's function using Riemann's theory?
In some instances, Riemman's theory of integration gives a finite value to some improper integrals, but Lebesgue theory does not. How do we know beforehand that Lebesgue theory does not give a different value than the Riemann's to some, say, a number-theoretic question? In which case, what does "the value of integral" mean?
Is it that once the theory is developed, it might have applications in the future? I am sorry my questions are incoherent and may be meaningless too, but I request to understand my difficulty in accepting two theories of integration in mathematics like one accepts Newton's theory of gravity and Einstein's theory of relativity in Physics. I urge you to resolve my difficulty in understanding.
If $f\colon[a,b]\longrightarrow\Bbb R$ is Riemann-integrable, then it is also Lebesgue-integrable, and the integrals are equal. So, in the context of the Lebesgue integration theory, there are more functions that you can integrate. Also, more natural statements about integrals are true in the context of Lebesgue integration. For instance, if $f$ is integrable and if $g\colon[a,b]\longrightarrow\Bbb R$ is such that $|g|\leqslant|f|$, then $g$ is integrable too (assuming that $g$ is measurable): this holds for the Lebesgue integral, but not for the Riemann one. Also, a natural distance from a function $f$ to a function $g$ in the context of a theory of integration is $\int_a^b|f(x)-g(x)|\,\mathrm dx$. With respect to this distance, the space of Riemann-integrable functions is not complete: there are Cauchy sequences that don't converge. But the space of Lebesgue-integrable functions is complete, and this is important for Analysis; it allows use to build new functions as limits of existing ones.