Proposition: Let $\mu$ be a borel measure such that $\mu(\Bbb{R^d})<+\infty$ Then $\mu$ is regular, i.e $\mu(A)=\inf\{\mu(G)|G \supset A, \text{ G open}\}=\sup\{\mu(K)|K \subseteq A, \text{ K compact}\}$ Using this ,i have to prove that if a Borel measure on $\Bbb{R^d}$ has the property ...

I'm trying to show that given two regular Borel measures on a compact space $X \subset \mathbb{R}$, $\mu_1, \mu_2$, and a bounded Borel function $f: X \to \mathbb{C}$, there exists a sequence $\{f_n\} \subset C(X)$ of continuous functions s.t $\int_X f_n d\mu_1 \underset{n \to \infty}{\to} \int_X...

Recently I noticed a tag "gre-exam" in this question. A quick search ([*-exam]) showed existence of similar tags for GMAT as well. In light of it, I wanted to ask the possibility of creating a tag csir-net-exam (or similar), as a lot of us aide to math.SE to prepare for it. I have already spotte...

I'm not seeing the point of gmat-exam; is there any good reason to keep this around?

There are 384 questions tagged gre-exam. This appears to be a meta tag that I cannot really imagine anyone watching or searching for. I think it should be deleted.

I'm studying for quals/prelims. Tests you take in your first year or two of PhD (at least in America). Sometimes I want to browse problems that specifically came up in some qualifying/preliminary exam. When I look for "measure theory" there are elementary or high level questions mixed in with th...