Pseudofinite groups and VC-dimension

We develop local NIP group theory in the context of pseudofinite groups. In particular, given a sufficiently saturated pseudofinite structure $G$ expanding a group, and left invariant NIP formula $\delta (x;\bar{y})$, we prove various aspects of “local fsg” for the right-stratified formula ${\delta }^r(x;\bar{y},u):=\delta (x\cdot u;\bar{y})$. This includes a ${\delta }^r$-type-definable connected component, uniqueness of the pseudofinite counting measure as a left-invariant measure on ${\delta }^r$-formulas, and generic compact domination for ${\delta }^r$-definable sets.