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Given a probability space $(X, {\cal B}, m)$, measure preservingtransformations $g_1, \dots , g_k$ of $X$, and a colour set $C$, a colouringrule 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 thefinitely $g_1 (x), \dots , g_k(x)$ with $g_i(x) \not= x$ for all $i$ and almostall $x$. 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 therule at $x$ if $f(x) \in F( x, f(g_1 x), \dots , f(g_k x))$. A colouring ruleis paradoxical if it can be satisfied in some way almost everywhere withrespect to $m$, but not in {\bf any} way that is measurable with respect to afinitely additive measure that extends the probability measure $m$ and forwhich the finitely many transformations $g_1, \dots , g_k$ remain measurepreserving. 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 thecolour set $C$ are compact sets, $C$ is convex and finite dimensional, and thecolouring rule says if $c: X\rightarrow C$ is the colouring function then thecolour $c(x)$ must lie ($m$ a.e.) in $F(x, c(g_1(x) ), \dots , c(g_k(x)))$ fora non-empty upper-semi-continuous convex-valued correspondence $F$ defined on$X\times C^k$. We show that any colouring that approximates the correspondenceby $\epsilon$ for small enough positive $\epsilon$ cannot be measurable in thesame finitely additive way. Furthermore any function satisfying the colouringrule illustrates a paradox through finitely many measure preserving shiftsdefining injective maps from the whole space to subsets of measure summing upto less than one.

A Continuous Paradoxical Colouring Rule Using Group Action