  We construct a family of measures for random fields based on the iterated
subdivision of simple geometric shapes (triangles, squares, tetrahedrons) into
a finite number of similar shapes. The intent is to construct continuum limits
of scale invariant scalar field theories, by imitating Wiener's construction of
the measure on the space of functions of one variable. These are Gaussian
measures, except for one example of a non-Gaussian fixed point for the Ising
model on a fractal. In the continuum limits what we construct have correlation
functions that vary as a power of distance. In most cases this is a positive
power (as for the Wiener measure) but we also find a few examples with negative
exponent. In all cases the exponent is an irrational number, which depends on
the particular subdivision scheme used. This suggests that the continuum limits
corresponds to quantum field theories (random fields) on spaces of fractional
dimension.
