We consider approximate solution sequences of the 2D incompressible Euler equations obtained by mollifying compactly supported initial vorticities in L-p, 1 less than or equal to p less than or equal to 2, or bounded measures in H-loc(-1) and exactly solving the equations. For these solution sequences we obtain uniform estimates on the evolution of the mass of vorticity and on the measure of the support of vorticity outside a ball of radius R. If the initial vorticity is in L-p, 1 less than or equal to p less than or equal to 2, these uniform estimates imply certain a priori estimates for weak solutions which are weak limits of these approximations. In the case of nonnegative vorticities, we obtain results that extend, in a natural way, the cubic-root growth of the diameter of the support of vorticity proved first by C. Marchioro for bounded initial vorticities [Comm. Math. Phys., 164 (1994), pp. 507-524] and extended by two of the authors to initial vorticities in L-p, p > 2.311134153