Let $g(x) \colon \mathbb{R}^n_+ \to \mathbb{R}_+$ be homogeneous of order 1, concave and smooth function. Using entrywise product notation $x \circ p = (x_1 p_1, ..., x_n p_n)$ one can write $$ \mu_{n}(p) = \int\limits_{\mathbb{R}^n_+} g^n(x\circ p) e^{-g(x\circ p)} \mu(dx) $$ for some Borel finite measure $\mu$. I'm interested in possibility of reconstruction of $\mu$ by a set of $\mu_{n}$. Help me please with literature or with some ideas if it is possible.