In measure theory, a complete measure is a measure in which every subset of every null set is measurable (having measure 0).
Why do they say "every" null set ... i thought there could be only one, unique null set --> no elements, empty set !! ... and then "every subset of every null set" !! ... a null set will have ony one subset always (itself, and nothing else).
So, what is the meaning of "every subset of every null set"
- A null set is not to be confused with the empty set... a null set is a measurable set (a set element of a sigma algebra which itself has a measure attached to it) that has measure zero. A countable set of points in the reals has Lebesgue measure zero for example. In fact, there's more subtleties about a null set than it being of measure zero, see the article about it.
Can someone improve the examples section by giving an example of a Borel set and a non-measurable subset of it, explaining why the Borel measure is not complete. —Preceding unsigned comment added by 220.127.116.11 (talk) 20:20, 7 July 2008 (UTC)
I don't know of a constructive proof, but I know of an existence proof, based on the Cantor function (the only use for that function that I know of...)
First, you have to start knowing that every positive measure set contains a nonmeasurable subset. Let K be the Cantor set, and let f be the Cantor function. Obviously, f(Kc) is the Cantor set, which has measure zero, so f(K) has measure one. We need a strictly monotonic function, so consider , which is obviously strictly monotonic, hence one-to-one, hence a homeomorphism. Now, has measure one. Let be non-measurable, and let . Because is injective, we have that , and so is a null set. However, if it were measurable, then would be measurable (here I use the fact that the preimage of a Borel set by a continuous function is measurable, since continuous functions are measurable; is the preimage of through the continuous function .)
Actually, I just regurgitated that mostly from memory, but in my half-drunken state, right now I can't determine if that's correct. I've edited the article, but if I got it wrong, just delete the whole thing. Loisel (talk) 04:27, 8 July 2008 (UTC)