  We produce a series of Central Limit Theorems (CLTs) associated to compact
metric measure spaces $(K,d,\eta)$, with $\eta$ a reasonable probability
measure. For the first CLT, we can ignore $\eta$ by isometrically embedding $K$
into ${\mathcal C}(K)$, the space of continuous functions on $K$ with the sup
norm, and then applying known CLTs for sample means on Banach spaces (Theorem
3.1). However, the sample mean makes no sense back on $K$, so using $\eta$ we
develop a CLT for the sample Fr\'echet mean (Corollary 4.1). This involves
working on the closed convex hull of the embedded image of $K$. To work in the
easier Hilbert space setting of $L^2(K,\eta)$, we have to modify the metric $d$
to a related metric $d_\eta$. We obtain an $L^2$-CLT for both the sample mean
and the sample Fr\'echet mean (Theorem 5.1), and we relate the Fr\'echet sample
and population means on the closed convex hull to the Fr\'echet means on the
image of $K$. Since the $L^2$ and $L^\infty$ norms play important roles, in
Section 6 we develop a metric-measure criterion relating $d$ and $\eta$ under
which all $L^p$ norms are equivalent.
