Abstract
The concept of the Riemann integral is well known in mathematical analysis. Its essential shortage is connected with the fact that functions integrable according to Riemann must have “not too many” discontinuity points (the exact sense of this statement is clarified in Theorem 3.2). Thus, one can easily construct an example of a bounded measurable function which is not Riemann integrable (for example, the Dirichlet function, i.e., the indicator of the set ℚ of rational numbers, is not Riemann integrable on any interval).KeywordsMeasurable FunctionSummable FunctionFinite MeasureLimit TransitionMeasurable Bounded FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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