  We prove that for some manifolds $M$ the set of robustly transitive partially
hyperbolic diffeomorphisms of $M$ with one-dimensional nonhyperbolic centre
direction contains a $C^1$-open and dense subset of diffeomorphisms with
nonhyperbolic measures which are ergodic, fully supported and have positive
entropy. To do so, we formulate abstract conditions sufficient for the
construction of an ergodic, fully supported measure $\mu$ which has positive
entropy and is such that for a continuous function $\phi\colon X\to\mathbb{R}$
the integral $\int\phi\,d\mu$ vanishes. The criterion is an extended version of
the control at any scale with a long and sparse tail technique coming from the
previous works.
