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For an interval $I \\subset \\mathbb{R}$, define $\\alpha_{\\mu,\\nu}(I) := \\mathbb{W}_{1}(\\mu_{I},\\nu_{I})$, the Wasserstein distance of normalised blow-ups of $\\mu$ and $\\nu$ restricted to $I$. Let $\\mathcal{S}_{\\nu}$ be the square function $$\\mathcal{S}^{2}_{\\nu}(\\mu) = \\sum_{I \\in \\mathcal{D}} \\alpha_{\\mu,\\nu}^{2}(I)\\chi_{I},$$ where $\\mathcal{D}$ is the family of dyadic intervals of side-length at most one. 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For an interval $I \\subset \\mathbb{R}$, define $\\alpha_{\\mu,\\nu}(I) := \\mathbb{W}_{1}(\\mu_{I},\\nu_{I})$, the Wasserstein distance of normalised blow-ups of $\\mu$ and $\\nu$ restricted to $I$. Let $\\mathcal{S}_{\\nu}$ be the square function $$\\mathcal{S}^{2}_{\\nu}(\\mu) = \\sum_{I \\in \\mathcal{D}} \\alpha_{\\mu,\\nu}^{2}(I)\\chi_{I},$$ where $\\mathcal{D}$ is the family of dyadic intervals of side-length at most one. 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