The weighted total-variation measure space is complete

The map $J:\sigma \mapsto w\sigma$ is linear. Multiplication by the positive function $w$ gives $\mathrm{\mid }w\sigma \mathrm{\mid }=w\mathrm{\mid }\sigma \mathrm{\mid }$, so [F1] yields $\mathrm{\parallel }J\sigma {\mathrm{\parallel }}_{TV}=\mathrm{\parallel }\sigma {\mathrm{\parallel }}_w$. It is onto because for any finite signed measure $\eta$, $\sigma =w^{-1}\eta$ satisfies $J\sigma =\eta$ and $\mathrm{\parallel }\sigma {\mathrm{\parallel }}_w=\int wd\mathrm{\mid }{w}^{-1}\eta \mathrm{\mid }=\mathrm{\parallel }\eta {\mathrm{\parallel }}_{TV}$.

Let $({\eta }_n)$ be Cauchy in total variation. Choose increasing indices $n_j$ with $\mathrm{\parallel }{\eta }_{n_{j+1}}-{\eta }_{n_j}{\mathrm{\parallel }}_{TV}\le 2^{-j}$, taking the least qualifying index at each step. For each Borel set $A$, the series ${\eta }_{n_1}(A)+{\sum }_{j\ge 1}({\eta }_{n_{j+1}}-{\eta }_{n_j})(A)$ converges absolutely, uniformly in $A$, because each summand is bounded by its total-variation norm. Define $\eta (A)$ by that sum. If $(A_k)$ are disjoint, the double series of absolute values is bounded by $\mathrm{\parallel }{\eta }_{n_1}{\mathrm{\parallel }}_{TV}+{\sum }_j\mathrm{\parallel }{\eta }_{n_{j+1}}-{\eta }_{n_j}{\mathrm{\parallel }}_{TV}$, so the two countable sums may be interchanged; hence $\eta$ is countably additive. The same tail bound gives $\mathrm{\parallel }\eta -{\eta }_{n_j}{\mathrm{\parallel }}_{TV}\le {\sum }_{r\ge j}{2}^{-r}\to 0$. Given $\epsilon >0$, choose $j$ so this is below $\epsilon \mathrm{/}2$ and the Cauchy property gives $\mathrm{\parallel }{\eta }_n-{\eta }_m{\mathrm{\parallel }}_{TV}<\epsilon \mathrm{/}2$ for all $n,m\ge n_j$. For any $n\ge n_j$, choose $n_k\ge n_j$ and pass to the limit in $\mathrm{\parallel }{\eta }_n-{\eta }_{n_k}{\mathrm{\parallel }}_{TV}<\epsilon \mathrm{/}2$; since ${\eta }_{n_k}\to \eta$, this gives $\mathrm{\parallel }{\eta }_n-\eta {\mathrm{\parallel }}_{TV}\le \epsilon \mathrm{/}2<\epsilon$. Thus finite signed measures are complete in total variation.

Pulling the limit $\eta$ back through the inverse isometry $w^{-1}$ gives a limit in ${\mathcal{M}}_w(E)$ for every Cauchy sequence there.