= Dual Haar measure
{title2=$dm_{\widehat G}$}
The dual Haar measure is the scale of <Haar measure> on the <Pontryagin dual group> compatible with <Fourier inversion on a locally compact abelian group>. It can be constructed without first assuming inversion: for a compactly supported test function $\psi$, choose a finite sum of convolution squares $p$ with $\widehat p>0$ on its support and define $J(\psi)=\int\psi/\widehat p\,d\mu_p$. The <Bochner consistency identity for convolution squares> makes this independent of the choice. Modulation by a character shows $J$ is invariant under dual translation; the <Riesz representation on compactly supported continuous functions> gives the measure. It satisfies $d\mu_p=\widehat p\,dm_{\widehat G}$ and scales reciprocally when $dm_G$ is rescaled.
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