Let $(\Omega, \mathcal{F}, \textbf{P})$ be a probability space and $A,B, A_i$ events in $\mathcal{F}$. Prove the following properties of every probability measure:
- Monotonicity: If $A \subseteq B$ then $\textbf{P}(A) \leq \textbf{P}(B)$.
- Sub-additivity: If $A \subseteq \bigcup_{i} A_i$ then $\textbf{P}(A) \leq \sum_{i} \textbf{P}(A_i)$.
- Continuity from Below: If $A_{i} \uparrow A$, then $\textbf{P}(A_i) \uparrow \textbf{P}(A)$.
- Continuity from Above: If $A_{i} \downarrow A$, then $\textbf{P}(A_i) \downarrow \textbf{P}(A)$.
For the first, one could consider $\textbf{P}(B-A)$? For the second, one could consider $\textbf{P}(A_{i}-A)$ over all $i$? It seems like the third and fourth properties follow from the first two. In particular, if $A_{1} \subseteq A_2 \subseteq \dots$ and $\bigcup_{i} A_i = A$, then $\textbf{P}(A_1) \leq \textbf{P}(A_2) \leq \dots$ and $\sum_{i} \textbf{P}(A_i) = \textbf{P}(A)$.