Let X be a G -space where G is a topological group. We show that X is G -finitistic iff the orbit space X / G is finitistic. This result allows us to answer a question raised in [5] asking for an equivariant characterization of a non-finitistic G -space where G is a compact Lie group. For an arbitrary compact group G a simple characterization of G -finitistic spaces has been obtained in terms of new notions of G -compactness and G -dimension.