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Measure and Integral1
Published online by Cambridge University Press: 31 October 2008
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1. It is now nearly half a century since H. Lebesgue, whose obituary the reader may have seen in Nature not so long ago, created his theory of the integral which since then has superseded in modern analysis the classical conception due to B. Riemann. It is, I think, regrettable that knowledge of the Lebesgue integral seems to be still largely confined to the research worker. There is nothing unduly abstract or unnatural in this theory, nor anything in the proofs which would be too difficult for a good honours student to grasp. If the aim of university education be the teaching of general ideas and methods rather than that of technicalities, then the modern notion of the integral should not be omitted from the mathematical syllabus. It is the purpose of this purely expository note to sketch the build up of both the Riemann and the Lebesgue integral on the common geometrical basis of “measure” and thus to make evident to the uninitiated reader the striking advantages of the new integral.
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- Copyright © Edinburgh Mathematical Society 1947
References
* The inner closed squares of which the first four are drawn in Figure 4 approach S from inside. The open square S is the sum of this sequence of closed squares. The dotted lines indicate a representation of S as a sum of contiguous intervals.
1 A closer approach is effected in this way. (8.4) holds also for the content.
1 Compare Titchmarsh, E. C., The Theory of Functions, 2nd Ed. (Oxford, 1939), Chapter X.Google Scholar
2 The limit always exists but need not be finite, if f(x) is measurable and non-negative.
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