Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that

(i)	$\mathfrak{L}^2 = \mathfrak{L}^2 (M \to \mathbb{C})$ is a D316: Set of P-integrable complex Borel functions on $M$

The complex Lebesgue inner product on $\mathfrak{L}^2$ is the D4881: Complex function \begin{equation} \mathfrak{L}^2 \times \mathfrak{L}^2 \to \mathbb{C}, \quad (f, g) \mapsto \int_X f \bar{g} \, d \mu \end{equation}