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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