Given a normal domain R of finite type over a mixed characteristic complete DVR or a perfect field of characteristic p, one can define the notion of a test ideal. In equal characteristic p, it is well known that test ideals are stable under small perturbations. The proof of this fact boils down to a fundamental result in Smith's thesis on tight closure, which states that there are no nonzero almost zero elements in the top local cohomology of R^+.In this talk, I will explain how to extend this result to the mixed characteristic setting, along with its applications to mixed characteristic test ideals. Time permitting, I will also outline the key ideas behind the proof. This is joint work in progress with Bhargav Bhatt and Linquan Ma.