Let $\mu_1$ and $\mu_2$ be two finite measures defined on $\sigma(\mathcal{F})$ such that, $\forall A \in \mathcal{F}$, $\mu_1(A)=\mu_2(A)$. Show that they must agree on $\sigma(\mathcal{F})$.
I attempted proof of this as follows:
let $\mathcal{M}= \lbrace A \in \sigma(\mathcal{F}) | \mu_1(A)=\mu_2(A) \rbrace$. If we show $\mathcal{M}$ is a monotone class, we are done by the Monotone Class Theorem.
To that end, let $\lbrace A_n \rbrace$ be a sequence of non-decreasing sets from $\mathcal{M}$. We "disjointify" the sets : Let $B_k=A_k- \bigcup_{j=1}^{k-1}A_j$. Note that $\lbrace B_i \rbrace$ are pairwise disjoint and $\bigcup_{j=1}^{\infty}A_j=\bigcup_{j=1}^{\infty}B_j$. Now, since $\mu_1$ and $\mu_2$ are countably additive, $$\mu_1(\bigcup\limits_{j=1}^{\infty}B_j)= \sum\limits_{j=1}^{\infty}\mu_1(B_j),$$ $$\mu_2(\bigcup\limits_{j=1}^{\infty}B_j)= \sum\limits_{j=1}^{\infty}\mu_2(B_j).$$ Because $B_i$ is in $\mathcal{F}$, we have $\mu_1(B_i)=\mu_2(B_i), \forall i \in \mathbb{N}$, and hence $$\mu_1(\bigcup\limits_{j=1}^{\infty}B_j)=\mu_2(\bigcup\limits_{j=1}^{\infty}B_j)$$ We conclude that $\mathcal{M}$ is closed under increasing union. Now, to show $\mathcal{M}$ is closed under decreasing limits of sets, Let $\lbrace C_n \rbrace$ be non-increasing sets from $\mathcal{M}.$ We want to show $\lim C_n = \bigcap_{n=1}^{\infty}C_n$ is in $\mathcal{M}$. For this, define $S_k=\mu_1(C_k)$ and $T_k=\mu_2(C_k)$ as sequences of real numbers. Then, $\lim_{k \rightarrow \infty}S_k=\lim_{k \rightarrow \infty}\mu_1(S_k)=\lim_{k \rightarrow \infty}\mu_1(T_k)=\mu_1(\bigcap_{n=1}^{\infty}C_n)=\mu_2(\bigcap_{n=1}^{\infty}C_n)$, using continuity from above of a finite measure. Thus, $\mathcal{M}$ is a monotone class of $\mathcal{F}$ and by monotone class theorem, $\mathcal{M}(\mathcal{F})=\sigma(\mathcal{F})$.
Does this look alright?