Let $(\Sigma, F, \mu)$ be a measure space and let $A\in F$. If $\mu(A)< \infty$, is it true that $|A|<\infty$ ?
This is a result which seems to make sense intuitively and I would like to use it as a part of a proof I am attempting but cannot seem to be able to prove (or find a counter example to) this lemma.
If it is not true - are there any conditions one could apply to make it true?
Note sure I understand the intent. If $\mu$ is any finite measure, then $\mu(A) < \infty$ by definition, even on an uncountably infinite $A$.
E.g. a normal probability measure $p$ would produce $p(\mathbb{R}) = 1$.