The measures $\mu^{+}$and $\mu^{-}$constructed above are called the positive and negative parts of $\mu$, respectively. The measure
$$
|\mu|=\mu^{+}+\mu^{-}
$$
is called the total variation of $\mu$. The quantity
$$
|\mu|=|\mu|(X)
$$
is called the variation or the variation norm of $\mu$.
The decomposition $\mu=\mu^{+}-\mu^{-}$is called the Jordan or Jordan-Hahn decomposition.
One should not confuse the measure $|\mu|$ with the set function $A \mapsto|\mu(A)|$, which, typically, is not additive (e.g., if $|\mu|>\mu(X)=0$ ).
We observe that the measures $\mu^{+}$and $\mu^{-}$have the following properties that could be taken for their definitions:
$$
\begin{aligned}
&\mu^{+}(A)=\sup {\mu(B): B \subset A, B \in \mathcal{A}} \
&\mu^{-}(A)=\sup {-\mu(B): B \subset A, B \in \mathcal{A}}
\end{aligned}
$$
MATH3969 COURSE NOTES ：
belongs to $\mathcal{A}^{n}$ and the function
$$
\left(x_{1}, \ldots, x_{n}\right) \mapsto \mu^{(n)}\left(A^{x_{1}, \ldots, x_{n}}\right)
$$
is measurable with respect to $\bigotimes_{i=1}^{n} \mathcal{A}{i}$. Denote by $B{1}^{k}$ the set of all points $x_{1}$ such that
$$
\mu^{(1)}\left(A_{k}^{x_{1}}\right)>\varepsilon / 2 .
$$
Then $B_{1}^{k} \in \mathcal{A}{1}$ and $\mu{1}\left(B_{1}^{k}\right)>\varepsilon / 2$, which follows by Fubini’s theorem for finite products and the inequality $\mu\left(A_{k}\right)>\epsilon$. Indeed, $A_{k}=C_{m} \times X_{m+1} \times \cdots$