Maybe yet another argument in favour favor for the Riemann integral: it is fairly easy to generalize it to integration of vector-valued functions. Here the Lebesgue theory becomes more tricky as soon as the target space is something beyond a Banach space. In many situations you want to integrate a not too badly behaved function, say a continuous one, with values in a rather complicated topological vector space (not only FrechetFréchet, but perhaps dual of FrechetFréchet, only sequentially complete, or something like that if you think of distributions). OK, I admit that this is not what you teach in first year calculus, but just be prepared :)
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Maybe yet another argument in favour for the Riemann integral: it is fairly easy to generalize it to integration of vector-valued functions. Here the Lebesgue theory becomes more tricky as soon as the target space is something beyond Banach. In many situations you want to integrate a not too badly behaved function, say continuous, with values in a rather complicated topological vector space (not only Frechet, but perhaps dual of Frechet, only sequentially complete, or something like that if you think of distributions). OK, I admit that this is not what you teach in first year calculus, but just be prepared :) |
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