After the seminar work of Bourgain and Brezis in 2004, 2007, the theory of Sobolev inequalities in L1 has been rapidly developed over the last two decades.  While a basic example of such results has been known since the 1970s - the Korn-Sobolev inequality of M.J. Strauss - these surprising inequalities in L1 seem to have been largely unnoticed, perhaps because of their complicated dependence on the vector structure of the differential operators (This is in contrast to the scalar case where only the gradient appears.).  The characterization of homogeneous vector differential operators which support such inequalities is due to Van Schaftingen, the so-called cancelation condition, which is somewhat mysterious.  This mystery served as the impetus for the speaker, in collaboration with Dmitriy Stolyarov, to introduce so-called dimension stable spaces of measures.  These spaces consist of scalar measures which absent the vector structure of canceling differential operators capture some of their essential features, for example various Sobolev inequalities and dimension estimates.  In this talk we discuss in more detail this motivation for these spaces, as well as to give their definition, examples, and applications.  If time permits, we mention a recent application obtained with Riju Basak concerning estimates for the wave equation with data in these spaces.