I need to show using monotone class theorem (MCT) that; If $\mathcal{F}$ is a field, $P_1,P_2$ are two probability measures on $\sigma(\mathcal{F})$, then if $P_1=P_2$ on $\mathcal{F}$ then $P_1=P_2$ on $\sigma(\mathcal{F})$.
I started by defining $\mathcal{L}=\{A\in\sigma(\mathcal{F})| P_1(A)=P_2(A)\}$. Clearly we have $\mathcal{F}\in\mathcal{L}$. I then try to show that $\mathcal{L}$ is a monotone class, and then by (MCT), we have $\mathcal{L}=\sigma{\mathcal{F}}$.
How to show that if $A_i\in\mathcal{L}$ such that $A_i\uparrow A$, then $A\in \mathcal{L}$? Here $A_i$'s are not disjoint!
Hint: if $A_i \uparrow A$ then $P(A_i) \uparrow P(A)$. Can you prove this? Write $A = \cup_{n=1}^\infty (A_n\setminus A_{n-1})$ and note that this is a disjoint union (set $A_0 = \emptyset$ if necessary).