The Lebesgue integral seems to have been a fundamentally new way of thinking about the integral. It's hard to prove the convergence theorems if you have the Riemann integral in mind. And when you learn it, I suppose there are probably many instances where you also have can give a new way computer a very ineffective definition of thinking something and ask that it prove theorems. Ask it to prove anything about (absolutely convergent or non-negative) seriesthe primes where you start with the converse of Wilson's theorem as the definition of a prime. Can the computer figure out that its definition is terrible? Can it figure out what a prime really "is"?
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The Lebesgue integral seems to have been a fundamentally new way of thinking about the integral. It's hard to prove the convergence theorems if you have the Riemann integral in mind. And when you learn it, you also have a new way of thinking about (absolutely convergent or non-negative) series. |
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