Given a probability measure $\mathbf{P}$ on the interval $I=[0,1]$, we get a corresponding function $f:(\mathbb{R}_{>0})^2 \rightarrow \mathbb{R}_{>0}$ as follows: $$f(x,y) = \int_\mathbf{P} x^q y^{1-q}dq$$
It follows that $$f(x,x) = \int_\mathbf{P} xdq = x \int_\mathbf{P} dq = x\cdot 1 =x.$$
Presumably, $f(x,x) = x$ isn't the only thing we can derive here.
Question. Does this integral transform have a name, and which functions $f$ arise from a probability measure in this way?
Notice that $$ f(x,y)=yM_{\mathbf P}(\log(x/y)), $$ where $$ M_{\mathbf P}(t):=\int_{[0,1]}e^{tq}\,\mathbf P(dq),\qquad t\in\Bbb R, $$ is the moment generating function of the probability measure $\mathbf P$. The function $M_{\mathbf P}$ is necessarily continuous, $\log M_{\mathbf P}$ is convex (by Holder's inequality), and $M_{\mathbf P}(0)=1$. The function $0\le s\mapsto M_{\mathbf P}(-s)$ is the Laplace transform of $\mathbf P$. There is a characterization of Laplace transforms of probability measures on the half-line $[0,\infty)$ in terms of the notion complete monotonicity (for which Feller's Volume II is a good source of information). I don't know off the top of my head what extra condition is needed to ensure that $\mathbf P$ is concentrated on $[0,1]$.