Theorem: Uniqueness of Measures

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:
:$\mu = \nu$

Proof:
Define, for all $n \in \N$, $\DD_n$ by:
:$\DD_n := \set {E \in \Sigma: \map \mu {G_n \cap E} = \map \nu {G_n \cap E} }$
Let us show that $\DD_n$ is a Dynkin system.
By Intersection with Subset is Subset, $G_n \cap X = G_n$, whence $(3)$ implies that $X \in \DD_n$.
Now, let $D \in \DD_n$. Then:
{{begin-eqn}}
{{eqn | l = \map \mu {G_n \cap \paren {X \setminus D} }
      | r = \map \mu {G_n \setminus D}
      | c = Intersection with Set Difference is Set Difference with Intersection
}}
{{eqn | r = \map \mu {G_n} - \map \mu {G_n \cap D}
      | c = $\map \mu {G_n} < +\infty$, Set Difference and Intersection form Partition, Measure is Finitely Additive Function
}}
{{eqn | r = \map \nu {G_n} - \map \nu {G_n \cap D}
      | c = $(3)$, $D \in \DD_n$
}}
{{eqn | r = \map \nu {G_n \cap \paren {X \setminus D} }
      | c = Above reasoning in opposite order
}}
{{end-eqn}}
Therefore, $X \setminus D \in \DD_n$.
Finally, let $\sequence {D_m}_{m \mathop \in \N}$ be a sequence of pairwise disjoint sets in $\DD_n$.
Then:
{{begin-eqn}}
{{eqn | l = \map \mu {G_n \cap \paren {\bigcup_{m \mathop \in \N} D_m} }
      | r = \map \mu {\bigcup_{m \mathop \in \N} \paren {G_n \cap D_m} }
      | c = Intersection Distributes over Union
}}
{{eqn | r = \sum_{m \mathop \in \N} \map \mu {G_n \cap D_m}
      | c = $\mu$ is a measure
}}
{{eqn | r = \sum_{m \mathop \in \N} \map \nu {G_n \cap D_m}
      | c = $D_m \in \DD_n$
}}
{{eqn | r = \map \nu {G_n \cap \paren {\bigcup_{m \mathop \in \N} D_m} }
      | c = Above reasoning in opposite order
}}
{{end-eqn}}
Therefore:
:$\ds \bigcup_{m \mathop \in \N} D_m \in \DD_n$.
Thus, we have shown that $\DD_n$ is a Dynkin system.
Combining $(1)$ and $(3)$, it follows that:
:$\forall n \in \N: \GG \subseteq \DD_n$
From $(1)$ and Dynkin System with Generator Closed under Intersection is Sigma-Algebra:
:$\map \delta \GG = \map \sigma \GG = \Sigma$
where $\delta$ denotes generated Dynkin system.
By definition of $\map \delta \GG$, this means:
:$\forall n \in \N: \map \delta \GG \subseteq \DD_n$
That is:
:$\forall n \in \N: \Sigma \subseteq \DD_n \subseteq \Sigma$
By definition of set equality:
:$\Sigma = \DD_n$ for all $n \in \N$
Thus, for all $n \in \N$ and $E \in \Sigma$:
:$\map \mu {G_n \cap E} = \map \nu {G_n \cap E}$
Now, from Set Intersection Preserves Subsets, $E_n := G_n \cap E$ defines an increasing sequence of sets with limit:
:$\ds \bigcup_{n \mathop \in \N} \paren {G_n \cap E} = \paren {\bigcup_{n \mathop \in \N} G_n} \cap E = X \cap E = E$
from Intersection Distributes over Union and Intersection with Subset is Subset.
Thus, for all $E \in \Sigma$:
{{begin-eqn}}
{{eqn | l = \map \mu E
      | r = \lim_{n \mathop \to \infty} \map \mu {G_n \cap E}
      | c = Characterization of Measures: $(3)$
}}
{{eqn | r = \lim_{n \mathop \to \infty} \map \nu {G_n \cap E}
      | c = $\forall n \mathop \in \N: E \in \DD_n$
}}
{{eqn | r = \map \nu E
      | c = Characterization of Measures: $(3)$
}}
{{end-eqn}}
That is to say, $\mu = \nu$.
{{qed}}