1. The problem asks to state the defining properties of a probability measure $P$ on a measurable space $(\Omega, \mathcal{F})$. 2. A probability measure $P$ is a function from the sigma-algebra $\mathcal{F}$ to the interval $[0,1]$ that satisfies the following axioms: 3. **Non-negativity:** For every event $A \in \mathcal{F}$, the probability is non-negative: $$P(A) \geq 0$$ 4. **Normalization:** The probability of the entire sample space $\Omega$ is 1: $$P(\Omega) = 1$$ 5. **Countable additivity (sigma-additivity):** For any countable sequence of pairwise disjoint events $A_1, A_2, A_3, \ldots \in \mathcal{F}$, the probability of their union is the sum of their probabilities: $$P\left(\bigcup_{i=1}^\infty A_i\right) = \sum_{i=1}^\infty P(A_i)$$ 6. These properties ensure that $P$ behaves like a measure and assigns probabilities consistently to events in $\mathcal{F}$. **Final answer:** A probability measure $P$ on $(\Omega, \mathcal{F})$ satisfies: $$\begin{cases} P(A) \geq 0 & \text{for all } A \in \mathcal{F} \\ P(\Omega) = 1 \\ P\left(\bigcup_{i=1}^\infty A_i\right) = \sum_{i=1}^\infty P(A_i) & \text{for disjoint } A_i \end{cases}$$