Q:

Finite Measure: Uncountably Many Masses?

Can a finite measure space have uncountably many disjoint sets with nonvanishing mass:
$$\mu(A_{\lambda})>0\quad(\lambda\in\Lambda:\#\Lambda>\aleph_0)$$
Do you have an example or can you give me a proof, please?
(My problem is that an argument involving uncountable sums cannot be applied directly.)

A:

The standard argument is to consider
$$
            \Lambda_{n} = \{ \lambda \in \Lambda : \mu(A_{\lambda})\ge 1/n \},\;\;\; n =1,2,3,\cdots .
$$
Because $\mu$ is a finite measure, then $\Lambda_{n}$ must be a finite (or empty) set. Then you argue that
$$
                           \Lambda = \bigcup_{n=1}^{\infty}\Lambda_{n}.
$$

