取一个概率三元组，表示一次抛硬币。$(\Omega,\mathscr{F},\mathbb{P})$

然后

$$\begin{align} & \Omega = \{H,T\}. \tag{Prop. 1} \\ \end{align}$$

现在，

$$\begin{align} \text{A } \sigma \text{-algebra must be closed under countable union}, \tag{Prop. 2} \end{align}$$

所以

$$\begin{align} \text{if we have } \{H\},\{T\} \in \mathscr{F} \text{ then we must also have } \{H,T\} \in \mathscr{F}, \tag{Prop. 3} \end{align}$$

但是

$$\begin{align} \{H,T\} \text{ is not a possible event resulting from a single coin toss.} \tag{Prop. 4} \end{align}$$

出了什么问题？