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

[Jonathan Cast]

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