一种利用群作用的连续Parcherical着色规则
A Continuous Paradoxical Colouring Rule Using Group Action
摘要：
给定概率空间$(X, {\cal B}, m)$、$X$的测度保持变换$g_1, \dots , g_k$和颜色集合$C$，着色规则是一种用$C$对空间着色的方法，使得点$x$所允许的颜色由该点的位置和具有$g_i(x) \not= x$的$g_1 (x), \dots , g_k(x)$ $g_1 (x), \dots , g_k(x)$的颜色确定，对于所有$i$和几乎所有$x$。我们将着色规则表示为在$X\times C^k$上定义的$F$与$C$中的值的对应关系。如果$f(x) \in F( x, f(g_1 x), \dots , f(g_k x))$，则函数$f: X\rightarrow C$满足$x$处的规则。一个着色规则是矛盾的，如果它可以以某种方式满足几乎在任何地方关于$m$，但不是以{ \bf any}的方式，这是可测量的关于一个可加性测度，扩展的概率测度$m$和其中的许多转换$g_1, \dots , g_k$保持测量。当$g_1, \dots, g_k$是一个群$G$的成员，概率空间$X$和颜色集$C$是紧集，$C$是凸的和有限维的，并且着色规则说如果$c: X\rightarrow C$是着色函数，那么颜色$c(x)$必须说谎（$m$ a.e.）在$F(x, c(g_1(x) ), \dots , c(g_k(x)))$中对于定义在$X\times C^k$上的非空上半连续凸值对应$F$.我们表明，任何着色，近似的对应$ε$足够小的积极$ε$不能在相同的可加性的方式测量。此外，任何函数满足着色规则说明了一个悖论，通过许多措施保持转移定义内射映射从整个空间的子集的措施总和小于1。
英文摘要：
Given a probability space $(X, {\cal B}, m)$, measure preserving transformations $g_1, \dots , g_k$ of $X$, and a colour set $C$, a colouring rule is a way to colour the space with $C$ such that the colours allowed for a point $x$ are determined by that point's location and the colours of the finitely $g_1 (x), \dots , g_k(x)$ with $g_i(x) \not= x$ for all $i$ and almost all $x$. We represent a colouring rule as a correspondence $F$ defined on $X\times C^k$ with values in $C$. A function $f: X\rightarrow C$ satisfies the rule at $x$ if $f(x) \in F( x, f(g_1 x), \dots , f(g_k x))$. A colouring rule is paradoxical if it can be satisfied in some way almost everywhere with respect to $m$, but not in {\bf any} way that is measurable with respect to a finitely additive measure that extends the probability measure $m$ and for which the finitely many transformations $g_1, \dots , g_k$ remain measure preserving. We show that a colouring rule can be paradoxical when the $g_1, \dots, g_k$ are members of a group $G$, the probability space $X$ and the colour set $C$ are compact sets, $C$ is convex and finite dimensional, and the colouring rule says if $c: X\rightarrow C$ is the colouring function then the colour $c(x)$ must lie ($m$ a.e.) in $F(x, c(g_1(x) ), \dots , c(g_k(x)))$ for a non-empty upper-semi-continuous convex-valued correspondence $F$ defined on $X\times C^k$. We show that any colouring that approximates the correspondence by $ε$ for small enough positive $ε$ cannot be measurable in the same finitely additive way. Furthermore any function satisfying the colouring rule illustrates a paradox through finitely many measure preserving shifts defining injective maps from the whole space to subsets of measure summing up to less than one.
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