Comment #8 on issue 1950 by asmeurer: Piecewise((x, 0 < x), (0, Eq(x, 0)), (-x, x < 0))._eval_interval(x, -1, 1) is wrong
http://code.google.com/p/sympy/issues/detail?id=1950
The Lebesgue Integral can be considered as a generalization of the Riemann Integral (even though they are completely different in form) because every Riemann integrable function is also Lebesgue integrable, and the integrals are the same. Therefore, it is often useful to intuitively think of integrals, even Riemann integrals, in terms of the Lebesgue theory, because then some concepts like the unimportance of sets of measure zero on the resulting integral.
And by the way, Riemann integration theory does not consider functions that take on values of +/-oo, but the Lebesgue theory does.
As for delta functions, this is something that I have not studied, but from what I understand, you have to use the DiracDelta function in SymPy to make it work correctly with integrate(). I think making Piecewise((oo, Eq(x, 0)), (0, True)) behave like the DiracDelta function is incorrect. That piecewise has an integral of 0 over the real line, whereas the DiracDelta function is defined a little differently so that it has an integral of 1. That is the reason that we have the DiracDelta function implemented. Actually, if you look at the rigorous definition on Wikipedia, it is defined by defining the Dirac delta as a different measure in the Lebesgue integral, though I think that just proves that the Lebesgue theory is everywhere.
Actually, I had it backwards. The result in 246 is correct (I was thinking of the integral of that function, but it's already been integrated).
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