Probability measures are required to satisfy
- $\mu(\emptyset) = 0$
- $\mu(\Omega) = 1$
- $\mu\left(\bigcup_{i \in \mathbb{N}} A_i \right) = \sum_{i\in \mathbb{N}} \mu(A_i)$ for disjoint sets $A_i \subset \Omega$.
What happens if we only assume the last condition for finite sequences? Do we lose (lots of or all) important results?