Re: [Haskell-cafe] Re: 0/0 > 1 == False

On 19 Jan 2008, at 2:52 AM, Kalman Noel wrote:
Jonathan Cast wrote:
On 12 Jan 2008, at 3:23 AM, Kalman Noel wrote:
(2) lim a_n = ∞ [...] (2) means that the sequence does not converge,
To a value in R. Again, inf is a perfectly well defined extended real number, and behaves like any other element of R u {-inf, inf}. (Although that structure isn't quite a field --- 0 * inf isn't defined, nor is inf - inf).
Out of curiosity, is there some typical application domain for extended real numbers?
In calculus and elementary analysis, the notion of limits at/to infinity, improper Riemann integrals, etc., are introduced by a succession of `special notations'. Taking the extended real numbers as the underlying space permits these notations to be defined more compositionally, because ∞ is now an ordinary mathematical object. For example, if f is a nonnegative measurable function, ∫f on a measurable set is /always/ defined (as an extended real number) and the special case of an `integrable' function is simply one where the integral (which is an actual mathematical value) is an element of R. So, when we say ∫f ∈ R, that notation is compositional --- that's real set membership there. Similarly, ∫_{-∞}^∞ f is defined (as a Lebesgue integral) the same way any other integral is, because the interval [-∞, ∞] is a perfectly good mathematical object. jcc
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Jonathan Cast