Are the dimensions of a set and its image equal under typical smooth functions?

We examine the question whether the dimension D of a set or probability measure is the same as the dimension of its image under a typical smooth function, if the range space is at least D-dimensional. If is a Borel probability measure of bounded support in Rn with correlation dimension D, and if m D, then under almost every continuously differentiable function (“almost every” in the sense of prevalence) from Rn to Rm, the correlation dimension of the image of is also D. If is the invariant measure of a dynamical system, the same is true for almost every delay coordinate map, under weak conditions on periodic orbits. That is, if m D, then m time delays are sufficient to find the correlation dimension using a typical measurement function. Further, it is shown that finite impulse response (FIR) filters do not change the correlation dimension. Analogous theorems hold for Hausdorff, pointwise, and information dimensions. We show by example that the conclusion fails for box-counting dimension.