\section{Preliminaries}
\label{sec:preliminaries}

\subsection{Notation}
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Let $\N$ be the set of natural numbers, $\R^n$ be an $n$-dimensional real Euclidean space, and $\R[x]$ be the set of polynomials in $x$ with real-valued coefficients. For a set $X \subseteq \R^n$, the sets $C(X)$ and $C_+(X)$ are respectively the set of continuous functions on $X$ and its nonnegative subcone. The subcone $C^1(X) \subset C(X)$ is composed of continuous functions on $X$ with continuous first derivatives. $\Mp{X}$ is the set of nonnegative Borel measures over $X$, and a duality pairing exists $\inp{f}{\mu} = \int_X f(x) d \mu(x)$ for all $f \in C(X), \ \mu \in \Mp{X}$. For every linear operator $\Lie$, there exists a unique linear adjoint $\Lie^\dagger$ such that $\inp{\Lie f}{\mu} = \inp{f}{\Lie^\dagger \mu}, \ \forall f,\mu$ is satisfied. An indicator function is $I_A(x)=1$ for a subset $A \subseteq X$ if $x \in A$ and $I_A(x) = 0$ otherwise. The measure of a set $A \subseteq X$ with respect to $\mu$ is $\mu(A) = \int_{A} d \mu =\int_X I_A(x) d\mu$. The quantity $\mu(X) = \inp{1}{\mu}$ is known as the `mass', and $\mu$ is a probability measure if $\mu(X) = 1$. The Dirac delta $\delta_{x'} \in \Mp{X}$ is a probability measure supported only on $x=x'$. For measures $\mu \in \Mp{X}, \ \nu \in \Mp{Y}$, the product measure satisfies $(\mu \otimes \nu)(A \times B) = \mu(A)\nu(B)$ for all $A \in X, \ B \in Y$. The projection map $\pi^x: X \times Y \rightarrow X$ returns only the $x$ coordinate $(x, y) \rightarrow x$. The pushforward by a function $f$ is the linear operator $f_\#$ satisfying $\inp{v(x)}{f_\# \mu} = \inp{v(f(x))}{\mu}$ for any test function $v \in C(X)$ and measure $\mu \in \Mp{X}$. The $x$-marginal of a measure $\mu \in \Mp{X \times Y}$ may be expressed as the pushforward of a projection $\pi_\#^x \mu$ with duality pairing $\inp{v(x)}{\pi_\#^x \mu} = \int_{X \times Y} v(x) d \mu(x, y) $ holding for all test functions $v(x) \in C(X)$.
