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 $$\|\mu\|_{TV} = \mu^+(X) + \mu^-(X)$$$$\|\mu\|_{TV} = \mu^+(X) + \mu^-(X) \label{0}\tag{0}$$ where $\mu^+$, $\mu^-$ is the Jordan-Hanh decomposition of $\mu$. Do we have the following properties (like in the case of probability measures):
$$
\|\mu\|_{TV} = \sup \limits_{A \in \mathcal{B}} \lbrace |\mu(A)| \rbrace .\label{1}\tag{1}$$
According to Bogachev p.177 vol. 1, \eqref{1} is not true if both measures $\mu^+$ and $\mu^-$ are nonzero, and only the following is valid:
$$
\|\mu\|_{TV} \leq 2 \sup \limits_{A \in \mathcal{B}} \lbrace |\mu(A)| \rbrace \leq 2 \|\mu\|_{TV}.
$$$$
\|\mu\|_{TV} \leq 2 \sup \limits_{A \in \mathcal{B}} \lbrace |\mu(A)| \rbrace \leq 2 \|\mu\|_{TV}. \label{2}\tag{2}
$$
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
$$ d_{TV}(P,Q):=\sup\limits_{A\in \mathcal B}|P(A) - Q(A)|,\label{3}\tag{3}$$
which seems to me to contradict Bogachev. Is there something I am misunderstanding ?
$$
\|\mu\|_{TV} = \frac{1}{2}\sup \limits_{f \text{msb},\|f\|_{\infty} \leq 1} \int f\ d\mu.\label{2}\tag{2}
$$$$
\|\mu\|_{TV} = \frac{1}{2}\sup \limits_{f \text{msb},\|f\|_{\infty} \leq 1} \int f\ d\mu.\label{4}\tag{4}
$$
I have the same question here, does property \eqref{24} holds for $\mu \in \mathcal{M}(X)$ ?
In addition, do you know of any reference treating these questions for totally finite measure (apart from Bogachev) ? Thanks !
