Recent content by hellbike

and for the second part?

let f_n be series of borel functions. Explain why set B = {x: \sum_n f_n(x) is not convergent} is borel set. Proof, that if\int_R |F_n|dY \leq 1/n^2 for every n then Y(B) = 0.Y is lebesgue measure.for first part i thought that set of A={x: convergent} is borel, and B=X\A so it's also borel...

I'm not sure if i understand this, because this seems too easy. It's obvious that Y(x(B)) = xY(B) is true for every interval. It's obvious that this set is closed under countable sums from definition of measure (this just requires construction of pair disjoint sets). A\B for every A,B from R is...

and this is for both, borel and measurable? Is showing that this is true for approximation using open intervals is enough for proving this for borel sets? because borel sets are these that can be approximated using open intervals, right? And what are measurable sets?

Show, that Y(x(B)) = xY(B) (Y is Lebesgue_measure ) for every borel set B and x>0. Show that also for measurable sets. I don't know how to prove anything for neither borelian or measurable sets, so I'm asking someone for doing this problem, so i can do other problems with borelian and...

yes, thank you very much.

What do you mean by counting method? Induction? And this sum is between 1/2 and 1, but i don't know why this would be useful? and for problem 1: -> is this sufficient to say that set of (p,q) for p,q in Q can do the job for any subset of R?

1.prove that for any set X: |X|<c <=> in P(X) exist such countable set family F, that sigma algebra generated by F contains all points. 2.let (X,E,u) be probability space and A_1,...,A_2009 in E have property u(A_i)>=1/2. Prove that there exist x such is in A_i for atleast 1005 different i. i...

I'm looking for book about Jordan measure and stuff like that.

\int_{0}^{1} (1-x^2)^n dx using binomial expansion: \int \sum_{k=0}^{n}( {n \choose k} (-x^2)^k) dx = \sum_{k=0}^{n}( {n \choose k} \frac{x (-x^2)^k)}{2 k+1}) and going to definite integral: \int_{0}^{1} (1-x^2)^n dx = \sum_{k=0}^{n}( {n \choose k} \frac{(-1)^k)}{2 k+1}) is this correct...

how to integrate (1-x^2)^n for n \in N ? Limits of integral are from 0 to 1, but i don't think that matter. (i tried to use latex for int, but it wasn't working).

f:(a,\infty)->R i want to prove, that, if function is convex, then: if exist x_1 \in R, exist x_2>x_1 : f(x_2)>f(x_1) then: for all x_3>x_2 for allx_4>x_3 : f(x_4)\ge f(x_3)\ge f(x_2) in other words: convex function is either decreasing on whole domain, or it starts to increase from some point...

What's difference between those Courant's books? They both seems to be first year calc books.

I'm looking for book about making proof. Is this kind of book even required to understand proofs? Is there some special theory behind proofs, or books about proofs just provide examples, and are more like "math for dummies" ? I'm not sure if it's proper to use that kind of book, should i...

I think that you'r over reatcing with that "you have chances near 0% to get academia position". maybe only 15% get academia position, but I'm sure not all PhDs want to work in academia. Some of them just made choise to work in industry. So you don't really compete with 100% of PhDs.