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Has infinite mathematics actually been useful for anything at all? I mean uncountable infinities, higher cardinalities, etc - not limits, and other 'pseudo-infinite' abstractions. I have never encountered these things applied to anything other than infinite mathematics for infinite mathematics' sake. Seems like a waste of time.

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Many of the ideas from infinite mathematics interact with finite mathematics in very interesting ways. For example, the traditional integral is the Riemann integral which cannot handle sets with too many (uncountably many) discontinuities. The Lebesgue integral loosens those restrictions by using measure to guarantee there aren't too many discontinuties. The Lebesgue integral, in turn, is used to integrate over function spaces, allowing for techniques like fourier transform and technologies such as mp3 encoding.

Why does encoding of discrete-time signals have to do with Lebesgue integrals?

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