  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 $\epsilon$ for small enough positive $\epsilon$ 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.
