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# Elements of Integration

DOI link for Elements of Integration

Elements of Integration book

# Elements of Integration

DOI link for Elements of Integration

Elements of Integration book

## ABSTRACT

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 was 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 C ? f ( t i ) X ( I i ) , where C T= IIi, is a finite decomposition of [a,b]

2 = 1 into subintervals, approach a unique llmiting value whenever the length of the largest interval goes to zero. A French mathematician, Henri Lebesgue (1875-1941), assumed that the above intervals Ii may be substituted by more general measurable sets and that the class of Riemann integrable functions can be enlarged to the class of measurable functions. In this case, we arrive a t a more solid theory of integration, which is better suited for dealing with various limit processes and which greatly contributed to the contemporary theory of probability and stochastic processes.