In this talk, we will provide a background on Gromov Hausdorff convergence including its measured version. Then we will consider metric spaces with doubling measures that support a Poincaré type inequality, which form an essential class of metric spaces in the study of Sobolev spaces, and present some important results due to Cheeger and Koskela. We will then present a method to approximate such a space, via maximal epsilon nets,while ensuring the persistence of a doubling measure and Poincaré equality. This is due to joint work with James Gill.