Consider the random variable $Y$ with finite support $\mathcal{Y}\equiv \{1,...,M\}$. Consider the random variable $X$ with support $\mathcal{X}$. Consider the random vector $V\equiv (V_1,...,V_M)$. Let $\text{Supp}(V|X=x)$ denote the support of $V$ conditional on $X=x$.
It is assumed that: for any $x\in \mathcal{X}$, the distribution of $V$ conditional on $X=x$ is absolutely continuous with respect to Lebesgue measure with everywhere positive density on $\text{Supp}(V|X=x)$.
Below, small case letters denote realisations of random variables/vectors.
Consider the following problem $$ (1) \hspace{1cm} \max_{y\in \mathcal{Y}} u_y(x)+v_y $$ where $u\equiv (u_1,...,u_M)$ is a vector of any $M$ functions of $X$.
Show that for any $x$, $v$, $u$, (1) has a unique solution almost surely. Could you help me with that? It should be related to the fact that $V$ conditional on $X=x$ is a continuous random variable and single realisations have probability mass equal to zero, but I am looking for a more formal argument.