In many contributions to the theory of Hausdorff measures, it has been the practice to place certain restrictions on the measure functions used. These restrictions usually ensure both the monotonicity and the continuity of the functions. The purpose of this work is to find conditions under which the restrictions of monotonicity and continuity may be relaxed. We shall be working mostly with sets in Euclidean space, although in the final theorem, we work in Hilbert space to show that some of our previous results do not generalize to this space. I am grateful to the referee for suggesting improvements in the proofs of the results.