= Solution
A measure $\mu$ on $(\Omega,\mathcal F)$ is $\sigma$-finite when
$$
\Omega=\bigcup_{n\geq1}E_n
\qquad\text{with}\qquad
\mu(E_n)<\infty.
$$
The $\sigma$-finite uniqueness theorem says that two measures agreeing on a generating $\pi$-system agree on the generated $\sigma$-algebra, provided the space is covered by countably many sets in that system having finite common measure. Equivalently, a $\sigma$-finite premeasure has at most one extension to the generated $\sigma$-algebra.
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