[see also: ensure]
Observe how the completeness of $L^2$ was used to guarantee the existence of $f$.
However, (5) is sufficient to guarantee invertibility in $A$.
We are guaranteed only one dense product for each $k$.
This guarantees that $f$ satisfies all our requirements.
......, which, by another theorem of Kimney's, is more than enough to guarantee that $P$ gives $A$ outer measure 1.