Show set $\{x\in\Omega : \{x\}\in\mathcal{A}\text{ and } \mu(\{x\})>0\}$ is countable

Question

Let $(\Omega,\mathcal{A},\mu)$ be a finite measure space.

Prove that the set $\{x\in\Omega : \{x\}\in\mathcal{A}\text{ and } \mu(\{x\})>0\}$ is countable.

I can't figure out where to start. Could someone help me out?

Answer 1

Let $A_i = \{x \in \Omega : \{x\} \in A , \mu\{x\} > \frac 1i\}$, for $i \in \mathbb N$.

Then, the desired set is $\cup_i A_i$ (why?).

Each $A_i$ is countable, in fact has less than $i \times \mu(\Omega)$ elements (why?), therefore the desired set must be countable.

Answer 2

I would consider the sets $A_\epsilon = \{ x\in\Omega: \{x\} \in A \text{ and} \mu(\{x\}) > \epsilon\}$. Then the set you are considering is a countable union of such sets. It remains to show that $A_\epsilon$ itself is countable. Try using the fact that the space has finite measure.