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It is well-known that the Lebesgue integral generalises the Riemann integral.However, as is also well-known but less frequently well-explained, thisgeneralisation alone is not the reason why the Lebesgue integral is importantand needs to be a part of the arsenal of any mathematician, pure or applied.Those who understand the correct reasons for the importance of the Lebesgueintegral realise there are at least two crucial differences between the Riemannand Lebesgue theories. One is the difference between the Dominated ConvergenceTheorem in the two theories, and another is the completeness of the normedvector spaces of integrable functions. Here topological interpretations areprovided for the differences in the Dominated Convergence Theorems, andexplicit counterexamples are given which illustrate the deficiencies of theRiemann integral. Also illustrated are the deleterious consequences of thedefects in the Riemann integral on Fourier transform theory if one restricts toRiemann integrable functions.

Should we fly in the Lebesgue-designed airplane? -- The correct defence of the Lebesgue integral