Properly showing that uncountable sum of measures finite nonzero sets is infinite, given they are subsets of $X$ of finite measure.

In a question where $(X,\sum ,\mu )$ is a finite measure space, I am asked to show that the set $\{x|\mu (\{x\}>0)\}$ is countable at most.

While intuitively this has to be true, because as long as it is countable any series of nonzero measures of sets must converge and this can happen for geometric sequences of measures, but for an uncountable sum, this "can't" be the case, but how to show that- I can't seem to understand.

Firstly, is this the way to solve it? Trying to reach a contradiction, or maybe uncountable sum of elements is generally meaningless as an expression? I'm not sure, what

In a question where $(X,\sum ,\mu )$ is a finite measure space, I am asked to show that the set $\{x|\mu (\{x\}>0)\}$ is countable at most.

While intuitively this has to be true, because as long as it is countable any series of nonzero measures of sets must converge and this can happen for geometric sequences of measures, but for an uncountable sum, this "can't" be the case, but how to show that- I can't seem to understand.

Firstly, is this the way to solve it? Trying to reach a contradiction, or maybe uncountable sum of elements is generally meaningless as an expression? I'm not sure, what