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These are answers submitted by jakubi

If there are (many?) numbers whose character as irrational or transcendental has not been proved yet, it seems to me not feasible to implement an explicit computational representations of these characteristic functions. Could it be possible an abstract implementation? Something like: declare A the set of irrational numbers, declare f(A) its characteristic function, integrate f(A), etc.
with(VectorCalculus): SetCoordinates(cartesian[x,y,z]): hemisphere:= Surface(,phi = 0 .. Pi/2,theta = 0 .. 2*Pi,coords=spherical[r,phi,theta])
Surface(,phi = 0 .. Pi/2,theta = 0 .. 2*Pi,coords=spherical[r,phi,theta]);
I am missing something, as I do not get the same output as you. Eg, with this input Surface(,phi = 0 .. Pi/2,theta = 0 .. 2*Pi,coords=spherical[r,phi,theta]); in Filtered HTML, I get the output of the following post.
As an aside, is it possible to display here the 2D math output of the function call Surface(,phi = 0 .. Pi/2,theta = 0 .. 2*Pi,coords=spherical[r,phi,theta]) as it is shown in a worksheet executing the above sequence of commands?
I understand that Maple int means Riemann integral (is there any implementation of the Lebesgue integral in Maple, or any other CAS?). So, in this context, I think that g(s) is restricted to be Riemann integrable.
This limit exist, I think, only in the distributional sense. In this sense, the gaussian functions converge, for sigma -> 0, to the Dirac delta distribution. The support of the Dirac delta is bounded (it is just a point). And the space of distributions of bounded support is the dual space of (test) complex functions over the reals that are infinitely differentiable and have any support. So, at least for any such V(t), this limit exists in the distributional sense. Apparently the answer of limit(J, sigma = 0); is based on this assumption, or it is somehow related, as a trace of this command shows that a series is being used:
{--> enter series/int, args = `+`(`/`(`*`(`/`(1, 2), `*`(V(`+`(mu, `-`(`*`(s,
`*`(`limit/X`))))), `*`(`^`(2, `/`(1, 2)), `*`(exp(`+`(`-`(`*`(`/`(1, 2),
`*`(`^`(s, 2)))))))))), `*`(`^`(Pi, `/`(1, 2))))), s = `+`(`-`(infinity)) ..
infinity, limit/X
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