Quoting @Lazy answering Set of atoms must be countable
If there are uncountably many atoms there has to be a set in any countable partition that contains uncountably many of these. But any uncountable set of singletons with positive mass has countable subsets with arbitrarily large measure. So the set cannot have finite measure.
In the definition of a measure it says $$ A \subset \bigcup_{k=1}^{\infty} A_k \implies \mu(A) ≤ \sum_{k=1}^{\infty} \mu(A_k) $$ and for subadditivity $$ \mu(\bigcup_{k=1}^{\infty} A_k) ≤ \sum_{k=1}^{\infty} \mu(A_k) $$
So even if any of these $A_k$s has infinite measure and hence $\sum_{k=1}^{\infty} \mu(A_k)=\infty$ that just means that $\mu(\bigcup_{k=1}^{\infty} A_k) ≤ \infty$ which could be anything.
I don't understand which concept I'm missing here. Could someone please explain in more detail?
