Exercise 3, § 2

Let X be a set, and $\mathscr{T}, \mathscr{T}'$ two Hausdorff topologies on X, with $\mathscr{T}'$ coarser than $\mathscr{T}$. Denote by T (resp. T') the set X equipped with the topology $\mathscr{T}$ (resp. $\mathscr{T}'$), and by j the identity mapping of T into T' (which is continuous). Assume that every bounded positive measure on T' is carried by the union of a sequence of compact subsets of T. For every topology $\mathscr{S}$ intermediate to $\mathscr{T}$ and $\mathscr{T}'$, denote by T_\mathscr{S} the set X equipped with the topology $\mathscr{S}$. Show that the set of bounded measures on T_\mathscr{S} is independent of $\mathscr{S}$.