I have to solve a tricky exercise that requires some ability to work with sets and random variables. I would like your help with this because I'm unable to see how to proceed.
Construction of $\mathcal{A},\mathcal{B}, \mathcal{C}$:
Consider $3$ sets $A,B,C$ with $A\equiv \{a_1,a_2,a_3\}$, $B\equiv \{b_1,b_2,b_3\}$, $C\equiv \{c_1,c_2,c_3\}$.
Let $\mathcal{A},\mathcal{B}, \mathcal{C}$ be the collection of all subsets of $A$,$B$,$C$ excluding the empty set, i.e., $$ \mathcal{A}\equiv \Big\{A, \{a_1\}, \{a_2\}, \{a_3\}, \{a_1,a_2\}, \{a_2, a_3\}, \{a_1,a_3\}\Big\} $$
$$ \mathcal{B}\equiv \Big\{B, \{b_1\},\{b_2\}, \{b_3\},\{b_1,b_2\}, \{b_2, b_3\}, \{b_1,b_3\}\Big\} $$
$$ \mathcal{C}\equiv \Big\{C, \{c_1\},\{c_2\}, \{c_3\},\{c_1,c_2\}, \{c_2, c_3\}, \{c_1,c_3\}\Big\} $$
Construction of $A\times B \times C$:
Let $\times$ denote the Cartesian Product. $A\times B \times C$ is the collection of all ordered $3$-tuples from $A,B,C$, i.e., $$ A\times B \times C\equiv \Big\{(a_1,b_1,c_1),(a_1,b_2,c_1), (a_1,b_3,c_1),... \Big\} $$
Construction of $\mathcal{T}$:
Consider $\mathcal{T}\equiv \mathcal{A}\times \mathcal{B}\times \mathcal{C}$. For example, an element $T\in \mathcal{T}$ is $$ T\equiv \Big(\{a_1,a_2\}, \{b_3\} , \{c_1,c_2\}\Big) $$
Each $T\in \mathcal{T}$ is composed of 3 "components". In the example above, the first component is $\{a_1,a_2\}$, the second is $\{b_3\}$, the third is $\{c_1,c_2\}$.
We rewrite each $T\in \mathcal{T}$ as the set of all possible ordered $3$-tuples that can be constructed from its $3$ components. Continuing the example above, $$ T\equiv \overbrace{\Big(\{a_1,a_2\}, \{b_3\} , \{c_1,c_2\}\Big)}^{\text{Representation 1}} \equiv \overbrace{\Big\{\overbrace{(a_1,b_3, c_1)}^{\text{ordered $3$-tuple}}, (a_2, b_3, c_1), (a_1,b_3, c_2), (a_2,b_3, c_2)\Big\} }^{\text{Representation 2}} $$
Construction of $\mathcal{M}$:
We collect in $\mathcal{M}$ all the subsets of $A\times B\times C$ that cannot be part of $\mathcal{T}$. For example, $$ M\equiv \Big\{(a_1,b_3, c_1), (a_2, b_2, c_1)\Big\} $$ $M$ is not an element of $\mathcal{T}$.
Probability: Suppose that I have a random vector $Y$ with size $3\times 1$ taking value in $A\times B\times C$.
Suppose that I have a random set $S$ taking value in $\mathcal{T}$.
Assume that $$ (\star)\hspace{1cm}\mathbb{P}(Y\in T)\geq \mathbb{P}(S\subseteq T ) \text{ } \forall T\in \mathcal{T} $$ where $\mathbb{P}$ denotes probability measure.
Question: show that $$ \mathbb{P}(Y\in M)\geq \mathbb{P}(S\subseteq M ) \text{ } \forall M\in \mathcal{M} $$
Some clarifications: Random sets are generalisations of random variables to sets.
I believe that to show what I want we should somehow use the fact that $\mathbb{P}(S=M )=0$ $\forall M\in \mathcal{M}$. But I can't see how to proceed.
Also, I realised that every set $M\in \mathcal{M}$ can be written as the union of all the subsets of $M$ that belong to $\mathcal{T}$. But I can't see where to use this.