Let (Mi,pi) be a sequence of n-dimensional Riemannian manifolds with lower Ricci curvature bound −1. Fix a non-orincipal ultrafilter and let X be the ultralimit of the sequence. Does there exists a p∈X and subsequence of (Mi,pi) converging to (X,p) in the pointed Gromov-Hausdorff sense? Answer AttributionSource : Link , Question Author : dg.jan , … Read more
Let U be a free ultrafilter on the positive integers N and fix U∈U such that U is not cofinite (thanks J.D.Hamkins for the correction.) Consider the natural bijection between (0,1] and the infinite {0,1}-sequences with infinitely many ones (written in base 2). Define also the sequence x=(xn) by xn=0 if n∈U and xn=1 otherwise, … Read more
If U is a selective (Ramsey) ultrafilter on ω and B is its base, then for every sequence A_0\supsetneq A_1\supsetneq A_2\supsetneq\ldots in \mathcal{B} there exists A\in\mathcal{B} such that |A\setminus A_n|\le n+1. I’m interested in the following generalization of this notion. Let \mathcal{B} be a base of a non-principal ultrafilter \mathcal{U} and let \mathcal{F}\subset[\omega]^\omega. Let us … Read more
A pre-ordered set is a pair (P,≤) where P is a set and ≤⊆P×P is a reflexive and transitive relation. Let NPU(ω) be the set of non-principal ultrafilters on ω. The Rudin-Keisler pre-order on NPU(ω) is defined by U≤RKV:⇔(∃f:ω→ω)(∀U∈U)f−1(U)∈V. It is easy to see that ≤RK is reflexive and transitive, but not anti-symmetric. Question. Given … Read more
Given a index set S together with a ultrafilter μ on S (such that no set of cardinality <|S| has measure 1). Let the ordered field R(S,μ) denote the ultrapower of R with respect to S, i.e. R(S,μ):=RS/∼, where two maps f,g are called equivalent, if μ({i∈S|f(i)=g(i)})=1. This is again an ordered field. Equip it … Read more
Let F be a free ultrafilter on N and, for each A⊆N and n∈N, define dn(A):=|A∩[1,n]|n. Question. Considering P(N)={0,1}N, is it true that the set {A⊆N:F–lim is not Borel? Answer Yes. Let me rather write subsets as sequences: the question is whether the subset L of (a_n) in \{0,1\}^{\mathbf{N}} such that \mathscr{F}\text{-}\lim\sum_{k=1}^n\frac{a_k}{n}=0 is non-Borel. Write … Read more
From the answer of Andreas Blass and comments of Ali Enayat on my question Selective ultrafilter and bijective mapping it became clear that for any free ultrafilter U we have U≁. Where \mathcal{F}\otimes\mathcal{G}=\{X\subset A\times B~|~\{a\in A~|~ \{b\in B~|~ (a,b)\in X\} \in\mathcal{G}\}\in\mathcal{F}\} for any two filters \mathcal{F}, \mathcal{G} on sets A, B respectively. And \mathcal{F}\sim\mathcal{G} means … Read more
Let X be a completely regular, Hausdorff topological space and let F be a z-ultrafilter on X. Then for each zero set W in X, either W∈F or there exists Z∈F such that Z does not meet W (this is the z-ultrafilter property). Now suppose that X is additionally normal. Then is it true that … Read more
Define a new product measure on cantor space as follows:u({0})=a,u({1})=1-a,where a∈(0,1/2]. Does any ultrafiter U hasn’t measure one? When a=1/2,I know U hasn’t measue one.I guess neither when a∈(0,1/2), but I don’t know how to prove. Answer This question is a bit more subtle than I had originally thought (in the comments), but anyway here’s … Read more
Consider two ultrafilters, $U$ and $V$, on the same cardinal $\kappa$. Let $D(U, V)=\lbrace X\subseteq \kappa: X\in U-V\rbrace$; clearly $D(U, V)$ is a lattice under $\subseteq, \cap, \cup $ since the intersection of two $U$- or $V$-large sets is $U$- or $V$-large, and the union of two $U$- or $V$-small sets is $U$- or $V$-small; … Read more
