Saharon Shelah
Miklos Laczkovich asked if there exists a Haussdorff (or even normal) space in which every subset is Borel yet it is not meager. The motivation of the last condition is that under MA_kappa every subspace of the reals of cardinality kappa has the property that all subsets are F_sigma, however Martin's axiom also implies that these subsets are meager. Here we answer Laczkovich' question.
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Let xi be a non-null countable ordinal. We study the Borel subsets of the plane that can be made $\bormxi$ by refining the Polish topology on the real line. These sets are called potentially $\bormxi$. We give a Hurewicz-like test to recognize potentially $\bormxi$ sets.
We give an affirmative answer to the following question: Is any Borel subset of a Cantor set $\textbf{ C}$ a sum of a countable number of pairwise disjoint $h$-homogeneous subspaces that are closed in $X$? It follows that every Borel set $X \subset \textbf{ R}^n$ can be partitioned into countably many $h$-homogeneous subspaces that are $G_δ$-sets in $X$.
The following will be shown: Let $I$ be a $σ$-ideal on a Polish space $X$ with the property that the associated forcing of $I^+$ Borel subsets ordered by $\subseteq$ is a proper forcing. Let E be an analytic or coanalytic equivalence relation on this Polish space with all equivalence classes Borel. If sharps of certain sets exist, then there is an $I^+$ Borel subset $C$ of $X$ such that $E \upharpoonright C$ is a Borel equivalence relation.
The following is true in the Solovay model. 1. If $\leq$ is a Borel partial quasi-order on a Borel set $D$ of the reals, $X$ is a ROD subset of $D$, and $\leq$ restricted to $X$ is linear, then $X$ is countably cofinal in the sense of $\leq$. 2. If in addition every countable set $Y$ of $D$ has a strict upper bound in the sense of $\leq$, then the ordering $< D ; \leq >$ has no maximal chains that are ROD sets.
On Borel sets in ideal topologies
本文研究了理想拓扑中Borel集的性质，证明了Borel层次不崩溃，且在特定条件下Borel集与分析集一致，并验证了近似引理在理想拓扑下的成立。
Define z to be the smallest cardinality of a function f:X->Y with X and Y sets of reals such that there is no Borel function g extending f. In this paper we prove that it is relatively consistent with ZFC to have b<z where b is, as usual, smallest cardinality of an unbounded family in w^w. This answers a question raised by Zapletal. We also show that it is relatively consistent with ZFC that there exists a set of reals X such that the Borel order of X is bounded but there exists a relatively analytic subset of X which is not relatively coanalytic. This answers a question of Mauldin.
We prove that it is relatively consistent with $\mathrm{ZFC}$ that every strong measure zero subset of the real line is meager-additive while there are uncountable strong measure zero sets (i.e., Borel's conjecture fails). This answers a long-standing question due to Bartoszyński and Judah.
The following is true in the Solovay model. 1. If $\le$ is a Borel partial order on a set $D$ of the reals, and $X$ is a ROD subset of $D$ linearly ordered by $\le$, then the restriction of $\le$ onto $X$ is countably cofinal. 2. If in addition every countable set $Y$ of $D$ has a strict upper bound in the sense of $\le$ then the ordering $< D ; \le >$ has no maximal chains that are ROD sets.
We characterize the completely determined Borel subsets of HYP as exactly the omega_1^{ck} subsets of HYP. As a result, HYP believes there is a Borel well-ordering of the reals, that the Borel Dual Ramsey Theorem fails, and that every Borel d-regular bipartite graph has a Borel perfect matching, among other examples. Therefore, the Borel Dual Ramsey Theorem and several theorems of descriptive combinatorics are not theories of hyperarithmetic analysis. In the case of the Borel Dual Ramsey Theorem, this answers a question of Astor, Dzhafarov, Montalban, Solomon & the third author.
We study the Borel subsets of the plane that can be made closed by refining the Polish topology on the real line. These sets are called potentially closed. We first compare Borel subsets of the plane using products of continuous functions. We show the existence of a perfect antichain made of minimal sets among non-potentially closed sets. We apply this result to graphs, quasi-orders and partial orders. We also give a non-potentially closed set minimum for another notion of comparison. Finally, we show that we cannot have injectivity in the Kechris-Solecki-Todorcevic dichotomy about analytic gr…
We study classes of Borel subsets of the real line $\mathbb{R}$ such as levels of the Borel hierarchy and the class of sets that are reducible to the set $\mathbb{Q}$ of rationals, endowed with the Wadge quasi-order of reducibility with respect to continuous functions on $\mathbb{R}$. Notably, we explore several structural properties of Borel subsets of $\mathbb{R}$ that diverge from those of Polish spaces with dimension zero. Our first main result is on the existence of embeddings of several posets into the restriction of this quasi-order to any Borel class that is strictly above the classes …
Every permutation invariant Borel subset of the space of countable structures is definable in $\La_{ω_1ω}$ by a theorem of Lopez-Escobar. We prove variants of this theorem relative to fixed relations and fixed non-permutation invariant quantifiers. Moreover we show that for every closed subgroup $G$ of the symmetric group $S_{\infty}$, there is a closed binary quantifier $Q$ such that the $G$-invariant subsets of the space of countable structures are exactly the $\La_{ω_1ω}(Q)$-definable sets.
Let $X$ be a Polish space and $K$ a separable compact subset of the first Baire class on $X$. For every sequence $\bs$ dense in $\kk$, the descriptive set-theoretic properties of the set \[ \lbf=\{L\in[\nn]: (f_n)_{n\in L} \text{is pointwise convergent}\} \] are analyzed. It is shown that if $K$ is not first countable, then $\lbf$ is $\PB^1_1$-complete. This can also happen even if $K$ is a pre-metric compactum of degree at most two, in the sense of S. Todorcevic. However, if $K$ is of degree exactly two, then $\lbf$ is always Borel. A deep result of G. Debs implies that $\lbf$ contains a Bore…
In this paper we complete the attempt of H. Lefmann to show that Borel equivalence relations on the $n$-element subsets of $2^ω$, that respect an order type, have a finite Ramsey basis.
On Borel subsets of generalized Baire spaces
本文在不假设 $κ^{<κ}=κ$ 的条件下发展广义 Baire 空间的描述集合论，修正基本拓扑概念，并应用于判定固定基数 $κ$ 的模型轨道是否为 $κ$-Borel，指出该问题取决于一阶理论的稳定性性质。
We call a subset S of a topological vector space V linearly Borel, if for every finite number n, the set of all linear combinations of S of length n is a Borel subset of V. It will be shown that a Hamel base of an infinite dimensional Banach space can never be linearly Borel. This answers a question of Anatolij Plichko.
We give a new and effective classification of all Borel Wadge classes of subsets of Baire space. This relies on the true stage machinery originally developed by Montalbán. We use this machinery to give a new proof of Louveau and Saint-Raymond's separation theorem for Borel Wadge classes. This gives a proof of Borel Wadge determinacy in the subsystem $\text{ATR}_0+Π^1_1$-I of second-order arithmetic.
Hausdorff dimension and countable Borel equivalence relations
本文证明可数 Borel 等价关系可在满 Hausdorff 维数的闭集上光滑化，并将结论推广到局部可数 Borel 拟序与低阶尺度函数情形。
Borel Reductions and Cub Games in Generalized Descriptive Set Theory
本文研究了在广义描述集合论中，Borel可约性如何将κ的幂集子集关系嵌入到2^κ上的Borel等价关系中。
We show that if $\mathcal{F}$ is any "well-behaved" subset of the Borel functions and we assume the Axiom of Determinacy then the hierarchy of degrees on $\pow(\mathbb{R})$ induced by $\mathcal{F}$ turns out to look like the Wadge hierarchy (which is the special case where $\mathcal{F}$ is the set of continuous functions).
A \emph{hull} of $A \subset [0,1]$ is a set $H$ containing $A$ such that $λ^*(H)=λ^*(A)$. We investigate all four versions of the following problem. Does there exist a monotone (wrt. inclusion) map that assigns a Borel/$G_δ$ hull to every negligible/measurable subset of $[0,1]$? Three versions turn out to be independent of ZFC (the usual Zermelo-Fraenkel axioms with the Axiom of Choice), while in the fourth case we only prove that the nonexistence of a monotone $G_δ$ hull operation for all measurable sets is consistent. It remains open whether existence here is also consistent. We also answer …
We show that for a $σ$-ideal $\ci$ with a Borel base of subsets of an uncountable Polish space, if $\ca$ is (in several senses) a "regular" family of subsets from $\ci $ then there is a subfamily of $\ca$ whose union is completely nonmeasurable i.e. its intersection with every Borel set not in $\ci $ does not belong to the smallest $σ$-algebra containing all Borel sets and $\ci.$ Our results generalize results from \cite{fourpoles} and \cite{fivepoles}.
Assume that there is no quasi-measurable cardinal smaller than $2^ω$. ($κ$ is quasi measurable if there exists $κ$-additive ideal $\ci $ of subsets of $κ$ such that the Boolean algebra $P(κ)/\ci$ satisfies c.c.c.) We show that for a c.c.c. $σ$-ideal I with a Borel base of subsets of an uncountable Polish space, if $\cal A$ is a point-finite family of subsets from I then there is an uncountable collection of pairwise disjoint subfamilies of $\cal A$ whose union is completely nonmeasurable i.e. its intersection with every non-small Borel set does not belong to the $σ$-field generated by Borel se…
We show that for every ordinal $α\in [1, ω_1)$ there is a closed set $F \subset 2^ω\times ω^ω$ such that for every $x \in 2^ω$ the section $\{y\in ω^ω; (x,y) \in F\}$ is a two-point set and $F$ cannot be covered by countably many graphs $B(n) \subset 2^ω\times ω^ω$ of functions of the variable $x \in 2^ω$ such that each $B(n)$ is in the additive Borel class $\boldsymbol Σ^0_α$. This rules out the possibility to have a quantitative version of the Luzin-Novikov theorem. The construction is a modification of the method of Harrington who invented it to show that there exists a countable $Π^0_1$ se…
For a cardinal lambda<lambda_{omega_1} we give a ccc forcing notion P which forces that for some Borel subset B of the Cantor space (1) there a sequence (eta_alpha:alpha<lambda) of distinct elements such that |(eta_alpha+B) cap (eta_beta+B)|>5 for all alpha,beta<lambda, but (2) there is no perfect set of such eta's. The construction closely follows the one from Shelah math.LO/9802134
Borel sets of Rado graphs and Ramsey's Theorem
本文针对 Rado 图这一齐性结构证明了 Galvin-Prikry 定理的无限维 Ramsey 类似结果，利用 Henson 图 Ramsey 理论技术和新的融合序列方法，克服了缺乏 Baire 空间融合性质的困难，为拓扑动力学对应关系的扩展奠定基础。
A subset $X$ of a Polish group $G$ is called \emph{Haar null} if there exists a Borel set $B \supset X$ and Borel probability measure $μ$ on $G$ such that $μ(gBh)=0$ for every $g,h \in G$. We prove that there exists a set $X \subset \mathbb{R}$ that is not Lebesgue null and a Borel probability measure $μ$ such that $μ(X + t) = 0$ for every $t \in \mathbb{R}$. This answers a question from David Fremlin's problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set $B$. (The answer was already known assuming the Continuum Hypothesis.) This resul…
Let $ξ\geq 1$ be a countable ordinal. We study the Borel subsets of the plane that can be made ${\bfΠ}^0_ξ$ by refining the Polish topology on the real line. These sets are called potentially ${\bfΠ}^0_ξ$. We give a Hurewicz-like test to recognize potentially ${\bfΠ}^0_ξ$ sets.
If $(X,d)$ is a Polish metric space of dimension $0$, then by Wadge's lemma, no more than two Borel subsets of $X$ can be incomparable with respect to continuous reducibility. In contrast, our main result shows that for any metric space $(X,d)$ of positive dimension, there are uncountably many Borel subsets of $(X,d)$ that are pairwise incomparable with respect to continuous reducibility. The reducibility that is given by the collection of continuous functions on a topological space $(X,τ)$ is called the \emph{Wadge quasi-order} for $(X,τ)$. We further show that this quasi-order, restricted to…
The generalized continuous model theory, Borel complexity and stability
本文研究了连续语言L对应的逻辑Iso(Y)-空间Y_L，探讨了博雷尔复杂性与稳定性分析的通用模型论框架。
Building on work of van Engelen and van Mill, we show that a zero-dimensional Borel space is homeomorphic to a semifilter if and only if it is homogeneous and not locally compact. Under $\mathbfΣ^1_1$-Determinacy, this result extends to all analytic and coanalytic spaces.
We give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to extend several prominent results in recursion theory (\eg the Shore-Slaman Join Theorem) to the setting of Polish spaces. As a by-product we give both positive and negative results on the Martin Conjecture on the degree preserving Borel functions between Polish spaces. Add…
Let $X$ be a zero-dimensional compact metrizable space endowed with a strictly positive continuous Borel $σ$-additive measure $μ$ which is good in the sense that for any clopen subsets $U,V\subset X$ with $μ(U)<μ(V)$ there is a clopen set $W\subset V$ with $μ(W)=μ(U)$. We study $σ$-ideals with Borel base on $X$ which are invariant under the action of the group $H_μ(X)$ of measure-preserving homeomorphisms of $(X,μ)$, and show that any such $σ$-ideal $\mathcal I$ is equal to one of seven $σ$-ideals: $\{\emptyset\}$, $[X]^{\leω}$, $\mathcal E$, $\mathcal M\cap\mathcal N$, $\mathcal M$, $\mathcal…
In $\mathsf{ZFC}$, if there is a measurable cardinal with infinitely many Woodin cardinals below it, then for every equivalence relation $E \in L(\mathbb{R})$ on $\mathbb{R}$ with all $\mathbfΔ_1^1$ classes and every $σ$-ideal $I$ on $\mathbb{R}$ so that the associated forcing $\mathbb{P}_I$ of $I^+$ $\mathbfΔ_1^1$ subsets is proper, there exists some $I^+$ $\mathbfΔ_1^1$ set $C$ so that $E \upharpoonright C$ is a $\mathbfΔ_1^1$ equivalence relation. In $\mathsf{ZF} + \mathsf{DC} + \mathsf{AD}_\mathbb{R} + V = L(\mathscr{P}(\mathbb{R}))$, for every equivalence relation $E$ on $\mathbb{R}$ with…
A Borel equivalence relation on a Polish space is said to be countable if all of its equivalence classes are countable. Standard examples of countable Borel equivalence relations (on the space of subsets of the integers) that occur in recursion theory are: recursive isomorphism, Turing equivalence, arithmetic equivalence, etc. There is a canonical hierarchy of complexity of countable Borel equivalence relations imposed by the notion of Borel reducibility. We will survey results and conjectures concerning the problem of identifying the place in this hierarchy of these equivalence relations from…
Reductions and necessary conditions for tall Borel Ramsey ideals
研究高博雷尔理想是否具拉姆齐性质这一开放问题，通过主要定理给出不存在可数局部解读等结论，还证明弱选择性$q^+$理想相关性质，提出博雷尔性是否强制可数解读的问题及后果，主要问题仍待解决。
CONTENTS OF THE ISSUE: Hurewicz-like tests for Borel subsets of the plane; Ordered Spaces, Metric Preimages, and Function Algebras; On the independence of a generalized statement of Egoroff's theorem from ZFC, after T. Weiss; Forty annotated questions about large topological groups; Strong compactness and a partition property; Countable Borel equivalence relations and quotient Borel spaces; Decisive creatures and large continuum; Models of real-valued measurability; Hausdorff ultrafilters; Block combinatorics; Maharam's problem; Universality of uniform Eberlein compacta; Linearly ordered compa…
Let $Γ$ be a countably infinite group. Given $k \in \mathbb{N}$, we use $\mathrm{Free}(k^Γ)$ to denote the free part of the Bernoulli shift action of $Γ$ on $k^Γ$. Seward and Tucker-Drob showed that there exists a free subshift $\mathcal{S} \subseteq \mathrm{Free}(2^Γ)$ such that every free Borel action of $Γ$ on a Polish space admits a Borel $Γ$-equivariant map to $\mathcal{S}$. Here we generalize this result as follows. Let $\mathcal{S}$ be a subshift of finite type (for example, $\mathcal{S}$ could be the set of all proper colorings of the Cayley graph of $Γ$ with some finite number of colo…
A nonsmooth path-connectedness relation in the real plane
通过构造实平面紧子集，证明其道路连通性关系与非光滑超有限 Borel 等价关系 Borel 双归约，从而回答既有公开问题。
We show that if $Y$ is one of the spaces $\ell^q$, $c_0$, $\ell^\infty$ or ${\textstyle \bigcap_{p > b}} \ell^p$ where $0 < q,b < \infty$, and the Fréchet space $\textstyle \bigcap_{p > a} \ell^p$ is contained in $Y$ properly, then $\textstyle \bigcap_{p > a} \ell^p$ first shows up in the Borel hierarchy of $Y$ at the multiplicative class of the third level. In particular $\textstyle \bigcap_{p > a} \ell^p$ is neither an $F_σ$ nor a $G_δ$ subset of $Y$. This answers a question by Nestoridis. This result provides a natural example of a set in the third level of the Borel hierarchy and with its …
We consider the reverse math strength of the statement $\mathsf{C\text-DM}$:"Every completely determined Borel set is measurable." Over $\mathsf{WWKL}_0$, we obtain the following results analogous to the previously studied category case. $\mathsf{C\text-DM}$ lies strictly between $\mathsf{ATR}_0$ and $\mathsf{L}_{ω_1,ω}\text-\mathsf{CA}$. Whenever $M\subseteq 2^ω$ is the second-order part of an $ω$-model of $\mathsf{C\text-DM}$, then for every $Z \in M$, there is a $R \in M$ such that $R$ is $Δ^1_1$-random relative to $Z$. On the other hand, without $\mathsf{WWKL}_0$, all sets have measure zer…
We show that every non-Haar-null analytic subset of $\mathbb{Z}^ω$ contains a non-Haar-null closed subset. Moreover, we also prove that the codes of Haar-null analytic subsets, and, consequently, closed Haar-null sets in the Effros Borel space of $\mathbb{Z}^ω$ form a $\mathbfΔ^1_2$ set.
In recent years much attention has been enjoyed by topological spaces which are dominated by second countable spaces. The origin of the concept dates back to the 1979 paper of Talagrand in which it was shown that for a compact space X, Cp(X) is dominated by P, the set of irrationals, if and only if Cp(X) is K-analytic. Cascales extended this result to spaces X which are angelic and finally in 2005 Tkachuk proved that the Talagrand result is true for all Tychnoff spaces X. In recent years, the notion of P-domination has enjoyed attention independent of Cp(X). In particular, Cascales, Orihuela a…
Suppose $E \subseteq \mathbb{R}$ is nowhere dense. If $(\mathbb{R},<,+,(x \mapsto λx)_{λ\in \mathbb{R} }, E)$ does not define every bounded Borel subset of every $\mathbb{R}^n$ then for every $s > 0$ we have $$ | \{ k \in \mathbb{Z}, -m \leq k \leq m - 1 : [k,k+1] \cap E \neq \emptyset \} | < m^s $$ for sufficiently large $m \in \mathbb{N}$. Then there is an $n \in \mathbb{N}$ and a linear $T : \mathbb{R}^n \to \mathbb{R}$ such that $T(E^n)$ is dense. It follows that if $E$ is in addition nowhere dense then $(\mathbb{R},<,+,0,(x \mapsto λx)_{λ\in \mathbb{R}}, E)$ defines every bounded Borel su…
The feeling that those two forcing notions-Cohen and Random-(equivalently the corresponding Boolean algebras Borel(R)/(meager sets), Borel(R)/(null sets)) are special, was probably old and widespread. A reasonable interpretation is to show them unique, or ``minimal'' or at least characteristic in a family of ``nice forcing'' like Borel. We shall interpret ``nice'' as Souslin as suggested by Judah Shelah [JdSh 292]. We divide the family of Souslin forcing to two, and expect that: among the first part, i.e. those adding some non-dominated real, Cohen is minimal (=is below every one), while among…
Partitioning the real line into Borel sets
本文研究实数线可被划分为多少个 Borel 集或闭集，证明除 Hausdorff 的 $\aleph_1$ 结果外，可能基数的谱在力迫扩张中可相当任意。
If $X$ is a set, $E$ is an equivalence relation on $X$, and $n \in ω$, then define $$[X]^n_E = \{(x_0, ..., x_{n - 1}) \in {}^nX : (\forall i,j)(i \neq j \Rightarrow \neg(x_i \ E \ x_j))\}.$$ For $n \in ω$, a set $X$ has the $n$-Jónsson property if and only if for every function $f : [X]^n_= \rightarrow X$, there exists some $Y \subseteq X$ with $X$ and $Y$ in bijection so that $f[[Y]^n_=] \neq X$. A set $X$ has the Jónsson property if and only for every function $f : (\bigcup_{n \in ω}[X]^n_=) \rightarrow X$, there exists some $Y \subseteq X$ with $X$ and $Y$ in bijection so that $f[\bigcup_{…
Let I be an ideal of subsets of a Polish space X, containing all singletons and possessing a Borel basis. Assuming that I does not satisfy ccc, we consider the following conditions (B), (M) and (D). Condition (B) states that there is a disjoint family F subseteq P(X) of size c, consisting of Borel sets which are not in I. Condition (M) states that there is a function f:X-> X with f^{-1}[{x}] notin I for each x in X. Provided that X is a group and I is invariant, condition (D) states that there exist a Borel set B notin I and a perfect set P subseteq X for which the family {B+x: x in P} is disj…
This paper answers three questions posed by the first author. In Theorem 2.6 we show that the family of strong measure zero subsets of {}^{omega_1}2 is 2^{aleph_1}-additive under GMA and CH. In Theorem 3.1 we prove that the generalized Borel conjecture is false in {}^{omega_1}2 assuming ZFC+CH. Next, in Theorem 4.2, we show that the family of subsets of {}^{omega_1}2 with the property of Baire is not closed under the Souslin operation.
Any free Borel flow is shown to admit a cross section with only two possible distances between adjacent points. Non smooth flows are proved to be Lebesgue orbit equivalent if and only if they admit the same number of invariant ergodic probability measures.
We show that a locally finite Borel graph is nonsmooth if and only if it admits marker sequences which are "far" from every point. Our proof uses the Galvin-Prikry theorem and the Glimm-Effros dichotomy.
A subset of a Polish space $X$ is called universally small if it belongs to each ccc $σ$-ideal with Borel base on $X$. Under CH in each uncountable Abelian Polish group $G$ we construct a universally small subset $A_0\subset G$ such that $|A_0\cap gA_0|=\mathfrak c$ for each $g\in G$. For each cardinal number $κ\in[5,\mathfrak c^+]$ the set $A_0$ contains a universally small subset $A$ of $G$ with sharp packing index $\pack^\sharp(A_κ)=\sup\{|\mathcal D|^+:\mathcal D\subset \{gA\}_{g\in G}$ is disjoint$\}$ equal to $κ$.
It is shown that Borel games of length $ω^2$ are determined if, and only if, for every countable ordinal $α$, there is a fine-structural, countably iterable extender model of Zermelo set theory with $α$-many iterated powersets above a limit of Woodin cardinals.
Given a space $X$ we investigate the descriptive complexity class $\G_X$ of the set $\FF_0(X)$ of all its closed zero-dimensional subsets, viewed as a subset of the hyperspace $\FF(X)$ of all closed subsets of $X$. We prove that $\max \{ \G_X; \ X \text{ analytic } \}=\pca $ and $\sup \{ \G_X; \ X \text{ Borel } \borm ξ\} \supseteq \Game \bora ξ$ for any countable ordinal $ξ\geq1$. In particular we prove that there exists a one-dimensional Polish subpace of $2^\wo\times \R^2$ for which $\FF_0(X)$ is not in the smallest non trivial pointclass closed under complementation and the Souslin operati…
We show that there exists a positive arithmetical formula $ψ(x,R)$, where $x \in ω$, $R \subseteq ω$, with no hyperarithmetical fixed point. This answers a question of Gerhard Jäger. As corollaries we obtain results on the proof-theoretic strength of the Kripke-Platek set theory; the fixed points of monotone functions in chain-complete partial orders; the non-Borel uniformization of Borel sets; and the hyperdegrees of fixed points of positive formulae. Further we prove a Suslin-Kleene type result for the specific encoding of the hyperarithmetical sets that we are using.
The measure half of the $2n+1$ problem
本文证明任意 Borel 函数生成的图在测度意义下具有最优的 $2n+1$ 着色，肯定回答了 Kechris 等提出的测度版本问题，且不依赖不变性或局部有限性。
Borel completeness of Tits buildings with no rank 3 residues of spherical type
本文研究了无球型3级残余的蒂茨建筑的博雷尔完备性，证明了其分类难度与可数图分类相当，并特别指出广义n边形的博雷尔完备性。
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