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Let $f(x)$ be a smooth function from $\mathbb{R}^n_{+}$ to $\mathbb{R}_{+}$ (it may be analytic). Are there some sufficient conditions on such function $f$ such that $$ f(x_1 y_1, \ldots, x_n y_n) = \int \limits_{\mathbb{R}^n_{+}} f(x_1 z_1, \ldots, x_n z_n) \mu (dz) $$ holds for probability measure $\mu$ concentrated in point $(y_1,\ldots,y_n)$ and doesn't hold for any other probability measure in $\mathbb{R}^{n}_{+}$?

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