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The characterizations are in terms of the functionals $\\nu_{\\gamma} (E_{\\lambda,\\gamma/p}[u])$ where \\[ E_{\\lambda,\\gamma/p}[u]= \\Big\\{(x,y )\\in \\mathbb{R}^N \\times \\mathbb{R}^N \\colon x \\neq y, \\, \\frac{|u(x)-u(y)|}{|x-y|^{1+\\gamma/p}}>\\lambda\\Big\\} \\] and the measure $\\nu_{\\gamma}$ is given by $\\mathrm{d} \\nu_\\gamma(x,y)=|x-y|^{\\gamma-N} \\mathrm{d} x \\mathrm{d} y$. 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The characterizations are in terms of the functionals $\\nu_{\\gamma} (E_{\\lambda,\\gamma/p}[u])$ where \\[ E_{\\lambda,\\gamma/p}[u]= \\Big\\{(x,y )\\in \\mathbb{R}^N \\times \\mathbb{R}^N \\colon x \\neq y, \\, \\frac{|u(x)-u(y)|}{|x-y|^{1+\\gamma/p}}>\\lambda\\Big\\} \\] and the measure $\\nu_{\\gamma}$ is given by $\\mathrm{d} \\nu_\\gamma(x,y)=|x-y|^{\\gamma-N} \\mathrm{d} x \\mathrm{d} y$. 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