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Definition D5632
Complex Lebesgue convolution approximate identity

Let $M = (\mathbb{R}^D, \mathcal{L}, \ell)$ be a D1744: Lebesgue measure space such that
 (i) $\eta_0, \eta_1, \eta_2, \ldots : \mathbb{R}^D \to \mathbb{C}$ are each an D1921: Absolutely integrable function on $M$ (ii) $\mathfrak{L}^p = \mathfrak{L}^p(M \to \mathbb{C})$ is the D316: Set of P-integrable complex Borel functions on $M$ for $p \in [1, \infty)$
Then $\eta = \{ \eta_n \}_{n \in \mathbb{N}}$ is an approximate identity for complex Lebesgue convolution on $M$ with respect to $p$ if and only if $$\forall \, f \in \mathfrak{L}^p : \lim_{n \to \infty} \Vert \eta_n * f - f \Vert_{L^p} = 0$$