  We explore the consequences for the boundedness properties of averaging and
maximal averaging operators, of the following local comparabiliity condition
for measures: Intersecting balls of the same radius have comparable sizes.
Since in geometrically doubling spaces this property yields the same results as
doubling, we study under which circumstances it is equivalent to the latter
condition, and when it is more general. We also study the concrete case of the
standard gaussian measure, where this property fails, but nevertheles averaging
operators are uniformly bounded, with respect to the radius, in $L^1$. However,
such bounds grow exponentially fast with the dimension, very much unlike the
case of Lebesgue measure.
