  We use persistent homology in order to define a family of fractal dimensions,
denoted $\mathrm{dim}_{\mathrm{PH}}^i(\mu)$ for each homological dimension
$i\ge 0$, assigned to a probability measure $\mu$ on a metric space. The case
of $0$-dimensional homology ($i=0$) relates to work by Michael J Steele (1988)
studying the total length of a minimal spanning tree on a random sampling of
points. Indeed, if $\mu$ is supported on a compact subset of Euclidean space
$\mathbb{R}^m$ for $m\ge2$, then Steele's work implies that
$\mathrm{dim}_{\mathrm{PH}}^0(\mu)=m$ if the absolutely continuous part of
$\mu$ has positive mass, and otherwise $\mathrm{dim}_{\mathrm{PH}}^0(\mu)<m$.
Experiments suggest that similar results may be true for higher-dimensional
homology $0<i<m$, though this is an open question. Our fractal dimension is
defined by considering a limit, as the number of points $n$ goes to infinity,
of the total sum of the $i$-dimensional persistent homology interval lengths
for $n$ random points selected from $\mu$ in an i.i.d. fashion. To some
measures $\mu,$ we are able to assign a finer invariant, a curve measuring the
limiting distribution of persistent homology interval lengths as the number of
points goes to infinity. We prove this limiting curve exists in the case of
$0$-dimensional homology when $\mu$ is the uniform distribution over the unit
interval, and conjecture that it exists when $\mu$ is the rescaled probability
measure for a compact set in Euclidean space with positive Lebesgue measure.
