Let $(X,\mathcal{M},\mu)$ be a Measure Space, then 1. $$\mu(\emptyset)=0$$ 2. Finite Additivity: $$\mu\left(\bigsqcup_{i=1}^{n}A_{i}\right)=\sum_{i=1}^{n}\mu(A_{i})\iff A_{i}\cap A_{j}=\emptyset,\ \forall i\not=j\text{, i.e. pairwise disjoint}$$ 3. Monotonicity: For $A,B\in\mathcal{M}$ $$A\subseteq B\implies \mu(A)\le\mu(B)$$ 4. If $(A_{n})_{n\in\mathbb{N}}\subset \mathcal{M}$, $A_{1}\subseteq A_{2}\subseteq\dots$ then $$\mu\left( \bigcup_{n=1}^{\infty}A_{n} \right)=\lim_{ n \to \infty } \mu(A_{n})$$ 5. If $(A_{n})_{n\in\mathbb{N}}\subset \mathcal{M}$, $A_{1}\supseteq A_{2}\supseteq\dots$ and $\exists k\ge 1$ s.t. $\mu(A_{k})<\infty$ then $$\mu\left( \bigcap_{n=1}^{\infty}A_{n} \right)=\lim_{ n \to \infty } \mu(A_{n})$$