If $\mu$ is a vector measure and $(A_n)_n$ is disjoint, are we able to show that $(\left\|\mu(A_n)\right\|)_n$ is summable$?

Question

Let

• $(\Omega,\mathcal A)$ be a measurable space
• $E$ be a $\mathbb R$-Banach space
• $\mu:\mathcal A\to E$ with $\mu(\emptyset)=0$ and $$\mu\left(\biguplus_{n\in\mathbb N}A_n\right)=\sum_{n\in\mathbb N}\mu(A_n)\tag1$$ for all disjoint $(A_n)_{n\in\mathbb N}\subseteq\mathcal A$

Let $(A_n)_{n\in\mathbb N}\subseteq\mathcal A$ be disjoint. Are we able to show that $(\left\|\mu(A_n)\right\|_E)_{n\in\mathbb N}$ is summable$?

Answer 1

No.

Say $(e_n)$ is an orthonomal sequence in a Hilbert space, and for $E\subset(0,\infty)$ define $$\mu(E)=\sum_{n\in E}e_n/n.$$It's not hard to show that $\mu$ is countably additive, but if $A_n=\{n\}$ then $\sum||\mu(A_n)||=\infty$.