Let $(X,d)$ be a metric space, $\mathcal{B}$ the Borel $\sigma$-algebra on $X$, and $\mathcal{M}(X)$ the space of totally finite measures on $\mathcal{B}$. Let $||\mu||_{TV}$ be the total variation norm on $\mathcal{M}(X)$ defined by $$||\mu||_{TV} = \mu^+(X) + \mu^-(X)$$ where $\mu^+$, $\mu^-$ is the Jordan-Hanh decomposition of $\mu$. Do we have the following properties (like in the case of probability measures):

(1) $||\mu||_{TV} = \sup \limits_{A \in \mathcal{B}} \lbrace |\mu(A)| \rbrace $.
According to Bogachev p.177 vol. 1, (1) is not true if both measures $\mu^+$ and $\mu^-$ are nonzero, and only the following is valid: $$||\mu||_{TV} \leq 2 \sup \limits_{A \in \mathcal{B}} \lbrace |\mu(A)| \rbrace \leq 2 ||\mu||_{TV}.$$ But I have seen (here for example) the total variation norm used as a metric (written $d_{TV}$) on the space of probability measure, with the following definition $$ d_{TV}(P,Q):=\sup\limits_{A\in \mathcal B}|P(A) - Q(A)|,$$ which seems to me to contradict Bogachev. Is there something I am misunderstanding ?

(2)$||\mu||_{TV} = \frac{1}{2}\sup \limits_{f \text{msb},||f||_{\infty} \leq 1} \int f\ d\mu.$

I have the same question here, does property (2) holds for $\mu \in \mathcal{M}(X)$ ?

In addition, do you know of any reference treating these questions for totally finite measure (appart from Bogachev) ? Thanks !