Proposition: Output-density identification

Let $f_A$ be the label-space test in XX-A and $f_D$ the test used with the background measure in XX-D. The two smeared functionals agree if

Keeping the same numerical test function while changing the integration measure does not in general keep the same smeared target.

Proof. XX-A uses $\mathrm{d}t{\mathrm{d}}^{3}x$. XX-D uses $\mathrm{d}{V}_{\bar{g}}=a(t{)}^3\mathrm{d}t{\mathrm{d}}^{3}x$. Equation (eq:density) makes their integrands and pairings equal. The multiplier is smooth and nowhere zero on $I$, so it preserves the compact test space. Without this conversion, the two pairings have different weights.