  A local Hausdorff dimension is defined on a metric space. We study its
properties and use it to define a local Hausdorff measure. We show that in the
case that in the local Hausdorff measure is finite we can recover the global
Hausdorff dimension from the local one. Lastly, for a variable Ahlfors
Q-regular measure on a compact metric space, we show the Ahlfors regular
measure is strongly equivalent to the local Hausdorff measure and that the
function $Q$ is equal to the local Hausdorff dimension.
