Let $X$ be a compact Hausdorff space. In chapter 3 of Peter Scholze's *Lectures on Analytic Geometry* he considers the space of signed Radon measures on $X$ equipped with the filtered colimit (aka inductive limit) topology of the (in the weak$^*$-topology) compact absolutely convex subsets $\mathcal{M}(X)_{\leq c}$. Here $\mathcal{M}(X)_{\leq c}$ denotes the subset of measures with total variation norm less or equal than $c$ with $c>0$.

Then he states that the resulting topology is a locally convex vector topology. I was wondering if the subsets $\mathcal{M}(X)_{\leq c}$ form a neighborhood basis of the origin. If the answer is yes, then I do not see why the resulting topology is not the same as the one induced by the total variation norm. If the answer is no, then I do not see how to show that this topology is a locally convex vector topology.

Any clarification on this would be really appreciated.