= Haar measure from normalized covering functionals
{c}
{title2=$I_u(f)=(f:u)/(h:u)$}
For nonzero nonnegative $u,h\in C_c(G)$, let $(f:u)$ be the infimum of $\sum_j a_j$ over finite dominations $f\leq\sum_j a_j u(x_j^{-1}\,\cdot)$ with $a_j>0$. Composition of covers gives $(f:u)\leq(f:h)(h:u)$, so $I_u(f)$ is uniformly bounded while $I_u(h)=1$. It is invariant under left translation, homogeneous, monotone and subadditive. As the support of $u$ shrinks to the identity, its additivity defect tends to zero: use the continuous proportions $f_i/(f_1+f_2+\delta p)$, with a cutoff $p$ positive on both supports, whose oscillations are small on every translate of that small support. In a locally compact sigma-compact metrizable group, diagonal convergence on a countable dense family, followed by uniform approximation on fixed compact supports, gives a positive invariant linear functional on all $C_c(G)$. The <Riesz representation on compactly supported continuous functions> turns it into <Haar measure>. Local compactness is essential for the compactly supported cutoffs and small-support test functions.
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