PomeB48EB7517450Proposition 18.26.9.B48EB7517450 The space $\prod _{\pi \in \mathscr R}\mathscr B_\pi $ is complete for the metric
The map $u\mapsto (\mu _\pi (u))_\pi $ takes values in this space and preserves the comparison $\delta _\omega ^{\mathrm{pol}}$.
Proof. The metric is finite, bounded by two, and separates points. A Cauchy sequence is Cauchy in total variation in each coordinate, and completeness of the space of finite signed measures gives a limit there; positivity and the mass bound pass to the limit. Given $\varepsilon >0$, choose $N$ such that every normalized coordinate distance is at most $\varepsilon $ for all $j,k\geq N$. Letting $k$ tend to infinity separately in each coordinate preserves this inequality, and taking the supremum proves convergence for $\widehat\delta $. The last assertion follows from the definition and the mass bound proved above. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.