  A length-$n$ random sequence $X_1,\ldots,X_n$ in a space $S$ is finitely
exchangeable if its distribution is invariant under all $n!$ permutations of
coordinates. Given $N > n$, we study the extendibility problem: when is it the
case that there is a length-$N$ exchangeable random sequence $Y_1,\ldots, Y_N$
so that $(Y_1,\ldots,Y_n)$ has the same distribution as $(X_1,\ldots,X_n)$? In
this paper, we give a necessary and sufficient condition so that, for given $n$
and $N$, the extendibility problem admits a solution. This is done by employing
functional-analytic and measure-theoretic arguments that take into account the
symmetry. We also address the problem of infinite extendibility. Our results
are valid when $X_1$ has a regular distribution in a locally compact Hausdorff
space $S$. We also revisit the problem of representation of the distribution of
a finitely exchangeable sequence.
