31,630 research outputs found
Disjoint Borel Functions
For each , we define a Borel function which encodes in a certain sense. We show that for each Borel
, implies where is any code for . We generalize this theorem for
in larger pointclasses . Specifically, if , then . Also for all , if
, then .Comment: 15 page
Disjoint Infinity-Borel Functions
This is a followup to a paper by the author where the disjointness relation
for definable functions from to is
analyzed. In that paper, for each we defined a Baire
class one function which encoded
in a certain sense. Given , let
be the statement that is disjoint from at most countably many of
the functions . We show the consistency strength of is that of an inaccessible cardinal. We show that
implies . Finally, we show that assuming large
cardinals, holds in models of the form
where is a selective ultrafilter on
.Comment: 16 page
Disjoint Infinity Borel Functions
Consider the statement that every uncountable set of reals can be surjected onto R by a Borel function. This is implied by the statement that every uncountable set of reals has a perfect subset. It is also implied by a new statement D which we will discuss: for each real a there is a Borel function fa : RtoR and for each function g : RtoR there is a countable set G(g) of reals such that the following is true: for each a in R and for each function g : R to R, if fa is disjoint from g, then a is in G(g). We will show that D follows from ZF +AD+ whereas the negation of D follows from ZFC
Menger curvature and rectifiability
For a Borel set E in R^n, the total Menger curvature of E, or c(E), is the
integral over E^3 (with respect to 1-dimensional Hausdorff measure in each
factor of E) of c(x,y,z)^2, where 1/c(x,y,z) is the radius of the circle
passing through three points x, y, and z in E.
Let H^1(X) denote the 1-dimensional Hausdorff measure of a set X. A Borel set
E in R^n is purely unrectifiable if for any Lipschitz function gamma from R to
R^n, H^1(E cap gamma(R)) = 0. It is said to be rectifiable if there exists a
countable family of Lipschitz functions gamma_i from R to R^n such that H^1(E -
union gamma_i(R)) = 0. It may be seen from this definition that any 1-set E
(that is, E Borel and 0<H^1(E)<\infty) can be decomposed into two disjoint
subsets E_irr and E_rect, where E_irr is purely unrectifiable and E_rect is
rectifiable.
Theorem. If E is a 1-set in R^n and c(E)^2 is finite, then E is rectifiable.Comment: 39 pages, 3 figures, published version, abstract added in migratio
On disjoint Borel uniformizations
Larman showed that any closed subset of the plane with uncountable vertical
cross-sections has aleph_1 disjoint Borel uniformizing sets. Here we show that
Larman's result is best possible: there exist closed sets with uncountable
cross-sections which do not have more than aleph_1 disjoint Borel
uniformizations, even if the continuum is much larger than aleph_1. This
negatively answers some questions of Mauldin. The proof is based on a result of
Stern, stating that certain Borel sets cannot be written as a small union of
low-level Borel sets. The proof of the latter result uses Steel's method of
forcing with tagged trees; a full presentation of this method, written in terms
of Baire category rather than forcing, is given here
Baire measurable paradoxical decompositions via matchings
We show that every locally finite bipartite Borel graph satisfying a
strengthening of Hall's condition has a Borel perfect matching on some comeager
invariant Borel set. We apply this to show that if a group acting by Borel
automorphisms on a Polish space has a paradoxical decomposition, then it admits
a paradoxical decomposition using pieces having the Baire property. This
strengthens a theorem of Dougherty and Foreman who showed that there is a
paradoxical decomposition of the unit ball in using Baire
measurable pieces. We also obtain a Baire category solution to the dynamical
von Neumann-Day problem: if is a nonamenable action of a group on a Polish
space by Borel automorphisms, then there is a free Baire measurable action
of on which is Lipschitz with respect to .Comment: Minor revision
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