论文标题
来自连续着色规则的量度理论悖论
A Measure Theoretic Paradox from a continuous colouring rule
论文作者
论文摘要
给定概率空间$（x，x，{\ cal b}，m）$，测量保留转换$ g_1，\ dots，\ dots，g_k $ $ x $和颜色套装$ c $，着色规则是一种与$ c $上色的方式，以$ c $为$ c $允许的apoint $ x $ x $确定，并由该点的位置和$ $ g_1（x）（x）（x）（x）（x）（x）（x）（x） $ g_i（x）\ not = x $用于所有$ i $，几乎所有$ x $。我们将着色规则表示为在$ x \ times c^k $上定义的信函$ f $，其中$ c $中的值。函数$ f：x \ rightArrow c $如果$ x $满足$ f（x）\在f（x，f（g_1 x），\ dots，f（g_k x））$中的规则。如果在$ m $方面几乎可以在某种程度上以某种方式满足着色规则，但在{\ bf ery}的方式上却不能以有限的附加度量来衡量，该方法扩展了$ {\ cal b} $定义的概率度量$ M $，并且为此，有一定的许多变换$ g_1 $ g_1 $ g_1 $ g_1 $ g_1，\ dots geptim \ dots geptive vervection，\ dots gepting beservect，保留了保留的保留。如果$ x $和颜色套装$ c $都是凸和紧凑的集合，并且着色规则说，如果$ c：x \ rightarrow c $是着色功能，那么着色规则是矛盾的，那么颜色$ c（x）$必须lie（$ m $ a.e. $ x \ times c^k $定义的上层 - 连续凸v值$ f $？答案是肯定的，我们提出了这样的例子。我们表明，此结果是强大的，包括任何近似$ε$的着色，对于$ε$，对于足够的正$ε$，也不能以相同的有限添加的方式来测量。由于可以通过连续函数近似欧几里得空间上的非空上音上音凸价值对应关系，因此存在连续函数定义的矛盾着色规则。
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 apoint $x$ 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 $m$ defined on ${\cal B}$ and for which the finitely many transformations $g_1, \dots , g_k$ remain measure preserving. Can a colouring rule be paradoxical if both $X$ and the colour set $C$ are convex and compact sets 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 $X\times C^k$? The answer is yes, and we present such an example. We show that this result is robust, including that any colouring that approximates the correspondence by $ε$ for small enough positive $ε$ also cannot be measurable in the same finitely additive way. Because non-empty upper-semi-continuous convex-valued correspondences on Euclidean space can be approximated by continuous functions, there are paradoxical colouring rules that are defined by continuous functions.
