ThmDex – An index of mathematical definitions, results, and conjectures.
Complex-linearity of real expectation

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $X, Y : \Omega \to \mathbb{R}$ are each a D3161: Random real number on $P$ (ii) $$\mathbb{E} |X|, \mathbb{E} |Y| < \infty$$ (iii) $\alpha, \beta \in \mathbb{C}$ are each a D1207: Complex number
Then $$\mathbb{E}(\alpha X + \beta Y) = \alpha \mathbb{E} X + \beta \mathbb{E} Y$$

Let $X, Y \in \text{Random}(\Omega \to \mathbb{R})$ each be a D3161: Random real number such that
 (i) $$\mathbb{E} |X|, \mathbb{E} |Y| < \infty$$
Let $\alpha, \beta \in \mathbb{C}$ each be a D1207: Complex number.
Then $$\mathbb{E}(\alpha X + \beta Y) = \alpha \mathbb{E} X + \beta \mathbb{E} Y$$
Proofs
Proof 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $X, Y : \Omega \to \mathbb{R}$ are each a D3161: Random real number on $P$ (ii) $$\mathbb{E} |X|, \mathbb{E} |Y| < \infty$$ (iii) $\alpha, \beta \in \mathbb{C}$ are each a D1207: Complex number
This result is a particular case of R1817: Complex-linearity of complex expectation. $\square$