The interplay projective measurements and strong correlations have emerged as a route to novel notions of phases and transitions. Recently, it was shown by Garratt et al that weak measurements on critical states can be cast into the framework of defect conformal field theory (CFT) and analyzed appropriately. In particular, they show that critical correlations of Luttinger liquids, described by the compact boson CFT can be modified by measurements when they are relevant (in the renormalization group sense) depending on the value of the Luttinger parameter. I will describe a generalization of this story by describing how different `symmetry enriched’ Luttinger liquid phases respond very differently for the same fixed measurement protocol. Some of these are identical to the story of Garratt et al while others differ either quantitatively, changing critical parameters, or qualitatively by being entirely immune to measurements. Altogether, I hope to show that measurements can richly modify ground state phase diagrams of strongly correlated systems. The talk will be based on upcoming work with Michele Fava and Sid Parameswaran.