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Definition D3983
Complex random Lebesgue inner product

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $\mathfrak{L}^2 = \mathfrak{L}^2 (P \to \mathbb{C})$ is a D3083: Set of P-integrable random complex numbers on $P$
The complex Lebesgue inner product on $\mathfrak{L}^2$ is the D4881: Complex function $$\mathfrak{L}^2 \times \mathfrak{L}^2 \to \mathbb{C}, \quad (X, Y) \mapsto \int_{\Omega} X \bar{Y} \, d \mathbb{P}$$

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $\mathfrak{L}^2 = \mathfrak{L}^2 (P \to \mathbb{C})$ is a D3083: Set of P-integrable random complex numbers on $P$
The complex Lebesgue inner product on $\mathfrak{L}^2$ is the D4881: Complex function $$\mathfrak{L}^2 \times \mathfrak{L}^2 \to \mathbb{C}, \quad (X, Y) \mapsto \mathbb{E}( X \bar{Y} )$$