Uniqueness of Measures/Proof 1

Theorem
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $\mathcal G$ be a generator for $\Sigma$.

Suppose that there exists a countable cover for $X$ by elements of $\mathcal G$ that have finite measure.

Let $\nu$ be a measure defined on $\Sigma$ such that $\mu \restriction_{\mathcal G} \, = \nu \restriction_{\mathcal G}$. Here, $\mu \restriction_{\mathcal G}$ denotes the restriction of $\mu$ to $\mathcal G$.

Then $\mu = \nu$ everywhere on $\Sigma$.