  We study parametrized families of orthogonal projections for which the
dimension of the parameter space is strictly less than that of the Grassmann
manifold. We answer the natural question of how much the Hausdorff dimension
may decrease by verifying the best possible lower bound for the dimension of
almost all projections of a finite measure. We also show that a similar result
is valid for smooth families of maps from $n$-dimensional Euclidean space to
$m$-dimensional one.
