  In this paper we show that, for topological dynamical systems with a dense
set (in the weak topology) of periodic measures, a typical (in Baire's sense)
invariant measure has, for each $q>0$, zero lower $q$-generalized fractal
dimension. This implies, in particular, that a typical invariant measure has
zero upper Hausdorff dimension and zero lower rate of recurrence. Of special
interest is the full-shift system $(X,T)$ (where $X= M^{\Z}$ is endowed with a
sub-exponential metric and the alphabet $M$ is a perfect and compact metric
space), for which we show that a typical invariant measure has, for each $q>1$,
infinite upper $q$-correlation dimension. Under the same conditions, we show
that a typical invariant measure has, for each $s\in(0,1)$ and each $q>1$, zero
lower $s$-generalized and infinite upper $q$-generalized dimensions.
