In this paper, we define spaces of measures DS beta([8d) with dimensional stability beta is an element of (0, d). These spaces bridge between Mb([8d), the space of finite Radon measures, and DSd([8d) = H1([8d), the real Hardy space. We show the spaces DS beta([8d) support Sobolev inequalities for beta is an element of (0, d], while for any beta is an element of [0, d] we show that the lower Hausdorff dimension of an element of DS beta([8d) is at least beta.