Proposition

For $A\in\mathcal{F}$: $$\mathbb{P}(A^{c})=1-\mathbb{P}(a)$$

Proposition (Monotonicity of Probability Measure)

For $A,B\in\mathcal{F}$, if $A\subseteq B$ then $$\mathbb{P}(A)\le \mathbb{P}(B)$$

Proposition (Inclusion-Exclusion)

For 2 events, let $A_1,A_2\in\mathcal{F}$. $$P(A_1\cup A_2)=P(A_1)+P(A_2)-P(A_1\cap A_2)$$ For $n$ events, let $A_1,...,A_n\in\mathcal{F}$: $$P(\bigcup^n_{i=1}A_i)=\sum^{n}_{k=1}(-1)^{k-1}\sum_{1\le i_1<\ldots< i_k\le n}P(A_{i_1}\cap \ldots\cap A_{i_k})$$

Proposition (Countable subadditivity)

For $(A_{n})_{n\ge 1}\subseteq \mathcal{F}$ we have that $$\mathbb{P}\left( \bigcup_{n\ge 1} A_{n} \right)\le \sum_{n\ge 1}\mathbb{P}(A_{n})$$