Prove that no finite measure space can have uncountably many members in its $\sigma$-algebra with strictly positive measure.

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Here is this week's POTW:

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Prove that no finite measure space can have uncountably many members in its $\sigma$-algebra with strictly positive measure.

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No one answered this week’s problem. You can read my solution below.

Spoiler

Suppose $(X,M,\mu)$ is a finite measure space with uncountably many $A\in M$ such that $\mu(A) > 0$. There exists an $n \in \Bbb N$ such that uncountably many $A\in M$ with $\mu(A) > 1/n$. If $A$ is the countable disjoint union of sets $A_i\in M$ of measure greater than $1/n$, then $A\in M$ and $\mu(A) = \sum \mu(A_i) = \infty$. This is a contradiction.