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