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In this paper, with a very simple argument, we develop a generalized version of the Mattila integral. Our first application is to consider the product of distances $$(\\Delta(E))^k= \\left\\{\\prod_{j=1}^k |x^j-y^j|: x^j, y^j\\in E\\right\\} $$ and show that when $d\\geq 2$, $(\\Delta(E))^k$ has positive Lebesgue measure if $\\dim_{\\mathcal{H}}(E)>\\frac{d}{2}+\\frac{1}{4k-1}$. 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