game-theoretic probabilities

Let $(\Omega ,\mathcal{Z})$ be a gamble space in game-theoretic probability. Given an event $A\subset \Omega$, we define the upper probability of $A$ as

$$\overline{P}(A)={\overline{\mathbb{E}}}^g[1_A],$$

where ${\overline{\mathbb{E}}}^g$ is the upper expectation (game-theoretic expectations). Similarly, the lower probability is defined as

$$\underline{P}(A)={\underline{\mathbb{E}}}^g[1_A].$$

If $\overline{P}(A)=\underline{P}(A)$ we simply write $P(A)$.