ThmDex – An index of mathematical definitions, results, and conjectures.
Real-linearity of complex expectation

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $Z, W : \Omega \to \mathbb{C}$ are each a D4877: Random complex number on $P$ (ii) $$\mathbb{E} |Z|, \mathbb{E} |W| < \infty$$ (iii) $\alpha, \beta \in \mathbb{R}$ are each a D993: Real number
Then $$\mathbb{E}(\alpha Z + \beta W) = \alpha \mathbb{E} Z + \beta \mathbb{E} W$$

Let $Z, W \in \text{Random}(\Omega \to \mathbb{C})$ each be a D4877: Random complex number such that
 (i) $$\mathbb{E} |Z|, \mathbb{E} |W| < \infty$$
Let $\alpha, \beta \in \mathbb{R}$ each be a D993: Real number.
Then $$\mathbb{E}(\alpha Z + \beta W) = \alpha \mathbb{E} Z + \beta \mathbb{E} W$$
Proofs
Proof 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $Z, W : \Omega \to \mathbb{C}$ are each a D4877: Random complex number on $P$ (ii) $$\mathbb{E} |Z|, \mathbb{E} |W| < \infty$$ (iii) $\alpha, \beta \in \mathbb{R}$ are each a D993: Real number
This result is a particular case of R1817: Complex-linearity of complex expectation. $\square$