[Haskell-cafe] Re: 0/0 > 1 == False
Jonathan Cast
jonathanccast at fastmail.fm
Sat Jan 19 19:05:57 EST 2008
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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