Uniqueness of Measures

Theorem
Let $\struct {X, \Sigma}$ be a measurable space.

Let $\GG \subseteq \powerset X$ be a generator for $\Sigma$; that is, $\Sigma = \map \sigma \GG$.

Suppose that $\GG$ satisfies the following conditions:


 * $(1):\quad \forall G, H \in \GG: G \cap H \in \GG$
 * $(2):\quad$ There exists an exhausting sequence $\sequence {G_n}_{n \mathop \in \N} \uparrow X$ in $\GG$

Let $\mu, \nu$ be measures on $\struct {X, \Sigma}$, and suppose that:


 * $(3):\quad \forall G \in \GG: \map \mu G = \map \nu G$
 * $(4):\quad \forall n \in \N: \map \mu {G_n}$ is finite

Then:
 * $\mu = \nu$

Alternatively, by Countable Cover induces Exhausting Sequence, the exhausting sequence in $(2)$ may be replaced by a countable $\GG$-cover $\sequence {G_n}_{n \mathop \in \N}$, still subject to $(4)$.