This is a linkpost for https://www.ykumar.org/almost-always/
I think you mean "The Lebesgue measure of the complement is zero" or something of the sort, not "The Lebesgue measure is non-zero". A set with nonzero measure has to be "fairly big" but it certainly doesn't amount to "almost all".
You can't say "almost all natural numbers are even" unless you have already picked a nonprincipal ultrafilter and it happens to have {even numbers} in it. Since there is no construction of a nonprincipal ultrafilter, what one always actually does is to say "Let U be a nonprincipal ultrafilter" and then proceed using just the properties that any nonprincipal ultrafilter has.
There's also "the complement is countable" when you're talking about real numbers, or more generally "the complement is of strictly smaller cardinality than the whole set". (When put like that, of course it's the same as "cofinite" for integers. Speaking of which, your heading for that one has "not finite" but again you mean (and say later on) "cofinite".)
Thanks for the advice! Still learning how to phrase things correctly & effectively.
I wasn't aware that you can't actually explicitly construct a nonprincipal ultrafilter - this is interesting and nonintuitive to me!
One proves the existence of nonprincipal ultrafilters with the axiom of choice. (You can show "every filter is contained in some ultrafilter" pretty directly using Zorn's lemma.) You don't actually need the full strength of AC -- I think "every infinite set has a nonprincipal ultrafilter" is weaker than countable choice, for instance -- but you definitely need some choice, and in particular "there are no nonprincipal ultrafilters" is known to be consistent with ZF.
A boring, colloquial way.
"Almost all the eggs are gone!" (as half a dozen remain)
A slightly-less-boring mathy way.
"Almost all prime numbers are odd!"
There are infinitely many primes. There is exactly one even prime number (2). Infinity minus one is... infinity.
"Almost all natural numbers are larger than one-hundred thousand quadrillion quadrillion vigintillion! (10^83)"
There are infinitely many natural numbers. There are infinitely many natural numbers larger than one-hundred thousand quadrillion quadrillion vigintillion. No practical difference between 10^83 and 1 (other than that one is an upper bound on the number of atoms in the universe).
You formalize this with sets: if you have some (infinite) set, a subset whose complement has a finite cardinality encompasses almost everything in the set. We call this cofiniteness.
A way to quantify surety.
"Almost all the people will lose money at the casino!"
I'm not sure what the exact rates are, but I'd bet money on this being true.
"Almost all numbers are composite!"
It is well known that the number of prime numbers below is approximately . In the limit as goes to infinity, the ratio of primes to non-primes numbers goes to approximately zero. So if you choose a random positive integer, I would bet my life savings that it's composite.
"Almost all graphs are asymmetric."
An intuitive explanation: if you take all possible combinations of nodes and vertices and let the number of each tend towards infinity, you should expect chaos to triumph over order. (sorry Ramsey). This does depend on a certain definition of symmetry, however, and a clearer statement would be "almost all graphs have only one automorphism."
A straightforward-yet-strange way for real numbers.
The Lebesgue measure is what you get when you try to generalize length, area, and volume to -dimensions. It forms the basis for our current understanding of integration, and helps us figure out how big stuff is.
Something with zero volume basically doesn't exist anyway.
"Almost all real numbers are irrational!"
Well yes, all the rationals have measure zero. All countable sets have measure zero, and the rationals are countable.
(Measures are nice: they're a neat generalization of physical scales (mass, volume, etc.) to arbitrary mathematical objects. They're particularly useful for dealing with continuous things we love Lebesgue measures)
"Almost all real numbers are noncomputable!"
Well yes? Of course an arbitrary real number can't be computed to arbitrary precision by a finite algorithm? Computable numbers are countable, remember? Nevermind that basically all the numbers we deal with are computable, this is obvious.
(Measure zero stuff basically doesn't exist, even if they're the only things we use on a daily basis)
"Almost all real numbers aren't in the Cantor set!"
Well yes, of course! Even though the Cantor set is uncountably infinite, it still has measure zero! It's a weird pseudo-fractal embedding of the real line that somehow manages to lose everything in translation but still keep all the relevant information.
(Idk, the Cantor set is weird)
A filter on an arbitrary set is a collection of subsets of that is closed under set intersections and supersets. (Note that this means that the smallest filter on is itself).
An ultrafilter is a filter which, for every , contains either or its complement. A principal ultrafilter contains a finite set.
A nonprincipal ultrafilter does not.
This turns out to be an incredibly powerful mathematical tool, and can be used to generalize the concept of "almost all" to esoteric mathematical objects that might not have well-defined or intuitive properties.
(One of the coolest uses of nonprincipal ultrafilters is in the construction of the hyperreals, post forthcoming).
Let be a nonprincipal ultrafilter over the natural numbers. It obviously contains no finite sets, but we run into a slight issue when we take the set
and its complement
By the filter axioms, only one of these can be in , and one of them has to be in . And thus, we can safely say:
"Almost all natural numbers are even."