Suppose I have a random variable $Y$ with support $\{1,2,..., M\}$.
Consider a random vector $V\equiv (V_1, V_2,..., V_M)$ with support $\mathcal{V}\subseteq \mathbb{R}^M$ with positive Lebesgue measure. By definition of support we know that $\mathcal{V}$ is a closed set (see here for example).
All random variables/vectors are defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
For any $j\in \{1,...,M\}$, let $$\mathcal{V}_j\equiv \{v\equiv (v_1,..., v_M)\in \mathcal{V} \text{ s.t. } v_j\geq v_k \text{ }\forall k\in \{1,...,M\} \text{ with } k\neq j\}$$ Hence, $\{\mathcal{V}_1, ..., \mathcal{V}_M\}$ constitutes a partition of $\mathcal{V}$.
Assumption A1: $V$ has a distribution absolutely continuous with respect to Lebesgue measure on $\mathcal{V}$
Assumption A2: $Y\in argmax_{k\in \{1,...,M\}} V_k$ with probability 1.
Question 1: Is it true that $\mathcal{V}_j$ is closed and convex?
Question 2: Under which sufficient conditions I can claim $$\mathbb{P}(Y=j)=\mathbb{P}(V\in \mathcal{V}_j)$$ $\forall j\in \{1,..., M\}$.
Attempted answer 1: we know that if $\mathcal{V}=\mathbb{R}^M$, then $\mathcal{V}_j$ is closed and convex (see here, for example). When $\mathcal{V}\subset \mathbb{R}^M$, I think (can you confirm?) we can generalise the same results for any closed $\mathcal{V}\subset \mathbb{R}^M$.
Attempted answer 2: by A1, I am tempted to naively set up the following proof:
wlog take $M=2$; for any $v\in \mathcal{V}$, suppose there exists $\{y,y'\}\subseteq argmax_{k\in \{1,2\}} v_k$ with $y\neq y'$. This can be the case if and only if $$ v_{y '}-v_y=0 $$ which is an event happening with probability measure zero, by A1, i.e., $$ \mathbb{P}(V \in \{v\in \mathcal{V} \text{ s.t. } v_1=v_2\} )=0 $$ Therefore, $argmax_{k} V_k$ is a singleton set almost surely.
This implies (not sure!) $$\mathbb{P}(Y=j)=\mathbb{P}(V\in \mathcal{V}_j)$$ $\forall j\in \{1,2\}$.
Doubts on the attempted answer 2: my attempted proof does not use the fact that $\mathcal{V}$ is closed or the fact that $\mathcal{V}_j$ is closed-convex.
However, I am confused by Theorem 1 in Lang (1986), which claims that under A1 and if $\mathcal{A}$ is a convex subset of $\mathcal{V}$, then $\mathbb{P}(V \in\partial \mathcal{A})=0$, where $\partial \mathcal{A}$ denotes the boundary of $\mathcal{A}$.
Any help to clarify? Is this Theorem relevant for my case? Referring to the case $M=2$, is $\partial \mathcal{V}_1\neq \{v\in \mathcal{V} \text{ s.t. } v_1=v_2\}$?