It's a bit strange that beth_1 isn't more used than it is, given the popularity of aleph_0. I mean people will rather supposed that their readers remember a special meaning for blackletter lowercase c...
Last week one of the guys here proved some theorem about partitions and whatnot, it was about finite subsets. He said that if we want to go to infinite sets then we need to talk about beth numbers instead of aleph (or assume GCH, of course)
I do not recall. It was something from a paper by some guy, the same title was used by Erdos and someone else (Rado?) 30 years later when they extended the results.
So @t.b., intuitively, that "A and B are necessarily Borel measurable" seems like it could be true, but the pessimistic analyst thinks that this is not the case.
@tb Hmm, I wonder what happens if there is a larger number of indicators mixed. Like a1, a_2, ..., a_n. Seems like there could be elaborate failure modes...
@TheChaz if a = b = 0 it's trivially wrong. If a = b you can take two non-measurable sets whose union is measurable (that's probably the one you did). If a = 0 then you can take A non-measurable.
@HenningMakholm Ooops, I forgot about that case too... Take two disjoint Borel sets A_1 and A_2 and split the complement into two non-measurable parts add one of them to both A_1 and A_2.
@anon (Sorry for replying to such an old chat message.) I'm surprised that someone with your math skills has never been to college. I presume you've taught yourself a lot since leaving high school? (Or was your high school particularly strong?)
@Mike: I took calc 3 / lin alg / diff equ in 11th grade from someone who should have really been teaching in uni - few liked him because he was rarely cognizant how over-everyone's-heads he was speaking (contra a few other math teachers, who were probably below a number of students' heads..) But most math I taught myself before seeing it in class - e.g. my dad gave me his college calc book when I was 12ish - so I served as sort of a bridge between teacher and classmates, educationally.
But yes, I read math in my free time. Nowadays I'm at the point I don't have enough time/concentration/motivation to do actually challenging study and non-challenging things don't really feel worth thinking about at all.
I was reading this thing about math education in high school (guy saying we don't need it) on reddit and everyone seemed to think it's was necessary to put everyone through the math classes
@anon Thanks for satisfying my curiosity. I'm impressed that you are as good as you are given that you are largely self-taught. I suspect you would match up well with some of the best senior math majors at the university where I teach.
He sure has his pet-peeves. Latex apart, how often have you guys seen the Morera thing, the cumulative distribution thing, and don't dare proving the infinitude of primes starting with the sentence let p_1,...,p_n be the n first primes and derive a contradiction.
I know we get frustrated sometimes with OP's, but I seem to have run into a string of nice ones recently. (See comments by dhz, fog, Carolus, and han.)
Calculating with Mathematica, one can have
$$\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}dt=\frac{\pi}{4}.$$
How can I get this formula by hand? Is there any simpler idea than using $u = \sin t$?
Is there a simple way to calculate
$$
\int_0^{\pi/2}\frac{\sin^n t}{\sin^n t+\cos^n t}dt
...
Calculating with Mathematica, one can have
$$\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}dt=\frac{\pi}{4}.$$
How can I get this formula by hand? Is there any simpler idea than using $u = \sin t$?
Is there a simple way to calculate
$$
\int_0^{\pi/2}\frac{\sin^n t}{\sin^n t+\cos^n t}dt
...
