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We study here a certain family of discrete measures $\\mu^r\\in\\mathcal P[0,N]$, coming from the idempotent state theory of $G$, which converge in Ces\\`aro limit to $\\mu$. Our main result is a duality formula of type $\\int_0^N(x/N)^pd\\mu^r(x)=\\int_0^N(x/N)^rd\\nu^p(x)$, where $\\mu^r,\\nu^r$ are the truncations of the spectral measures $\\mu,\\nu$ associated to $H,H^t$. 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