# Why does the supremum over finite partitions not suffice in defining total variation of complex measure?

In Rudin's Real and Complex Analysis, Chapter 6, eqn. 3, the total variation of a complex measure is defined as a supremum over all possible partitions of a set.

Why do we need to consider all possible partitions in the definition? Why isn't it enough to consider finite partitions?

Note that this is also the first exercise problem of Chapter 6 in Rudin.

Rudin asks you to decide whether it is enough, and indeed it is! Let $\mu$ be a complex measure on $(\Omega, \Sigma)$ and $$\lambda(E) := \sup_{\substack{E_1, \ldots, E_n \in \Sigma \\ \text{ partition of } E}} {\sum_{i=1}^n {|\mu(E_i)|}} \quad \forall\, E \in \Sigma.$$ Obviously, $\lambda \leq |\mu|$. Note that for any countable, measurable partition $(E_i)_i$ of $E$, $\sum_i {|\mu(E_i)|}$ converges, so it can be well approximated by finitely many terms. With an $(\varepsilon \to 0)$-argument this gives the other inequality.