Integration in Abstract Spaces

The historical significance of the development of measure theory is that it created a base for a generalization of the classical Riemann notion of the definite integral (which since 1854 has been considered to be the most general theory of integration). Riemann defined a bounded function over an interval [a, b] to be integrable if and only if the Darboux (or Cauchy) sums $$\sum^n_{i=1}f(t_i)\lambda(I_i)$$ where $$\sum^n_{i=1}(I_i)$$ is a finite decomposition of [a, b] into subintervals, approach a unique limiting value whenever the length of the largest subinterval goes to zero. The French mathematician, Henri Lebesgue (1875–1941), assumed that the above subintervals $$(I_i)$$ may be substituted by more general measurable sets and, in addition, that the class of Riemann integrable functions can be enlarged to the class of measurable functions. In this case, we arrive at a more advanced theory of integration, which is better suited for dealing with various limit processes and which led to the contemporary theory of probability and stochastic processes.