Let $M$ be a locally compact space and $\mu$ be some probability measure on $M$. Let $y^\ast \in M$, $f(x,y)$ be a real continuous bounded function $M \times M \to \mathbb{R}$. Consider an equality $$ f(x,y^\ast) = \int f(x,y) \, \mu(dy) $$ that holds for any $x \in M$. I want to find conditions when it is possible only for $\mu = \delta_{y^\ast}$.

If for any $x', x''$ there exists such $x'''$ that $f(x',y)f(x'',y) = f(x''',y)$ holds for any $y$ we can use the Stone-Weierstrass theorem to show that $\mu = \delta_{x^\ast}$. If this condition doesn't hold, we can try to apply some appropriate transform to our equality and to receive a new equality for which this condition holds. But it is art.

Consider an example. Let $M = \mathbb{R}^{n}_{+}$ and $f(x,y) = e^{-q(x \circ y)}$ where $q$ is positive, concave and positively homogeneous of order one function and $\circ$ is the entrywise multiplication. We have $$ e^{-q(x\circ y^\ast)} = \int_{\mathbb{R}^n_{+}} e^{-q(x \circ y)} \, \mu(dy) $$ If $q(x) = x_1 + \cdots + x_n$ we can apply the Stone-Weierstrass theorem. Hence functions of type $x \mapsto e^{-\langle x, y \rangle}$ are extremal for the set of the Laplace transforms of probability measures on $\mathbb{R}^{n}_{+}$. But for general function $q$ we can't be sure that we can apply this theorem. For example, we can't apply the Stone-Weierstrass theorem for function $q(x) = \min(x_1,\ldots,x_n)$.

I'm interested in other techniques that allow to say if $\mu = \delta_{y^\ast}$ or there are some other measures for which the equality is still true.


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