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Maybe this example helps: $$\Omega=\{\omega_1,\omega_2\}$$ Consider the following two probability measures $$\mathbb{P_1}(\{ \omega_1\}) = 0.4\ \ \ ; \ \ \mathbb{P_2}(\{ \omega_1\}) = 0.6 \$$ and $\mathbb{P_i}(\{ \omega_2\})=1-\mathbb{P_i}(\{ \omega_1\})$ Consider the two random variables:  X_1: \Omega \to \{0,1 \}, X_1(\omega_1) \mapsto 0 , X_1(\...