Weakly measurable seminorms and sufficient topologies

In this paper for $F$-space $\Gamma$ with an unconditional basis, embeddable in $L^0$, a sufficient topology stronger than that known earlier is constructed in the space $\Gamma^*$. In the case when $\Gamma$ has a sign-invariant basis it is shown that the topology on $\Gamma^*$ generated by weakly measurable seminorms ($w$-measurable in the sense of Gross seminorms according to Mushtari and Chuprunov) is sufficient.