Continuous covariant model and measurable completion

For two covariant sections, [F1] gives $\mathrm{\parallel }{F}_1(xh)-F_2(xh)\mathrm{\parallel }=\mathrm{\parallel }{F}_1(x)-F_2(x)\mathrm{\parallel }$. Thus their difference norm descends continuously to $G\mathrm{/}H$. Compact support modulo $H$ and Radon finiteness on compact sets make its square integrable, so the stated norm is well-defined.

Let $(F_n)$ be Cauchy in this norm. Choose a subsequence $(F_{n_k})$ such that ${\sum }_k\mathrm{\parallel }{F}_{n_{k+1}}-F_{n_k}{\mathrm{\parallel }}_2<\mathrm{\infty }$. The partial sums $S_m(q)={\sum }_{k<m}\mathrm{\parallel }{F}_{n_{k+1}}(x)-F_{n_k}(x)\mathrm{\parallel }$, with $q=xH$, satisfy $\mathrm{\parallel }{S}_m{\mathrm{\parallel }}_2\le {\sum }_k\mathrm{\parallel }{F}_{n_{k+1}}-F_{n_k}{\mathrm{\parallel }}_2$ by Minkowski. By [F3] and monotone convergence, $S={\lim }_m{S}_m$ is finite almost everywhere and has finite $L^2$ norm. Its infinite-value set is a measurable null set in $G\mathrm{/}H$. Outside its saturated preimage, the sections are pointwise Cauchy for every lift and converge to a covariant function $F$; set $F=0$ on the null fibers. On each compact neighborhood in $G$, the continuous approximants have jointly separable range, so their pointwise limit is locally strongly measurable. The $L^2$ norm of the tail is bounded by the tail of the same summable series, again by Minkowski and monotone convergence. Thus the embedded $L^2$ classes converge to $F$, which represents the original completion vector.

Equivalent Cauchy sequences have difference norm zero and so have representatives equal almost everywhere. Conversely, representatives equal almost everywhere have zero difference norm, hence define the same completion vector. This proves the asserted identification. ∎