A mathematical theory with deals with defining measures (some kind of volume) on arbitrary sets and using them for the definition of integrals.

The problem arises with the weaknesses of the Riemann integral.
E.g. in advanced probability theory you have to use integrals to define the mean of a random experiment, etc. But the set of elementary events over which you have the integrate can be an abstract, in fact very sick, set. So the Riemann integral won't be sufficient here, and you have develop new integration methods.

In the end you get the Lebesque integral which is much more powerful than the Riemann integral and it fixes most problems of the Riemann integral, e.g. you can integrate f, with f(x) = 0 for x in Q , f(x)=1 otherwise.

The foundations of this theory were created by Lebesque and Borel, both from Belgium (or the Netherlands?).