Thursday, December 14, 2023

Why we use the Lebesgue Integral

Non-mathematicians are often baffled at why mathematicians seek generalizations and abstractions, and often think that the abstractions can have no practical purpose.

Here is an example. The Riemann integral appears to suffice for any function of practical interest, and yet mathematicians insist on defining a Lebesgue integral to handle a wider variety of cases.

Andrew D. Lewis writes:

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

It is well-known that the Lebesgue integral generalises the Riemann integral. However, as is also well-known but less frequently well-explained, this generalisation alone is not the reason why the Lebesgue integral is important and needs to be a part of the arsenal of any mathematician, pure or applied. ...

The title of this paper is a reference to the well-known quote of the applied mathematician and engineer Richard W. Hamming (1915–1998):

Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.
The paper goes on to explain this very well. In particular, it shows why the Riemann integral is not good enough. In short, the Lebesgue integral makes Lp(R) complete normed vector spaces. If a sequence appears to converge, then it really does converge to a function in the space. This allows the use of limits in Fourier analysis, differential equations, and other areas of analysis.

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