Let $E$ be a countably infinite subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. We denote by $\mathbb{E}(X)$ the expectation of a real random variable $X$. Let $\mathscr{P}(E)$ be the set of subsets of $E$ and $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}$. Show that for all random variables $X$ and $Y$ on $(\Omega, \mathscr{A}, P)$ and for every subset $A$ of $E$: $$|\mu_X(A) - \mu_Y(A)| \leqslant \mathbb{E}\left(|\mathbb{1}_{\{X \in A\}} - \mathbb{1}_{\{Y \in A\}}|\right)$$ and deduce that $\|\mu_X - \mu_Y\| \leqslant P(X \neq Y)$, where $\{X \neq Y\} = \{\omega \in \Omega \text{ such that } X(\omega) \neq Y(\omega)\}$.
Let $E$ be a countably infinite subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. We denote by $\mathbb{E}(X)$ the expectation of a real random variable $X$. Let $\mathscr{P}(E)$ be the set of subsets of $E$ and $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}$.
Show that for all random variables $X$ and $Y$ on $(\Omega, \mathscr{A}, P)$ and for every subset $A$ of $E$:
$$|\mu_X(A) - \mu_Y(A)| \leqslant \mathbb{E}\left(|\mathbb{1}_{\{X \in A\}} - \mathbb{1}_{\{Y \in A\}}|\right)$$
and deduce that $\|\mu_X - \mu_Y\| \leqslant P(X \neq Y)$, where $\{X \neq Y\} = \{\omega \in \Omega \text{ such that } X(\omega) \neq Y(\omega)\}$.