Let $\mathcal{A}$ be a semiring on a set $\Omega$. And let $\mu$ be a measure on $\sigma(\mathcal{A})$ that is $\sigma$-finite on $\mathcal{A}$. Prove that there exists a sequence of sets in $\sigma(\mathcal{A})$, with finite measure, that increases to $\Omega$.
This has been used to prove a part of the Approximation Theorem for Measures in Achim Klenke's book on Probability Theory.
The measure is $\sigma$-finite only on $\mathcal{A}$ but we are choosing sets of finite measure from $\sigma(\mathcal{A})$, and this sequence increases to $\Omega$. I do not see how this can be possible. We can not choose the trivial example of the sequence being just $\{\Omega\}$ as $\Omega$ need not belong to an arbitrary semiring, further $\Omega$ may not have finite measure.