ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Deduction system
Zermelo-Fraenkel set theory
Binary cartesian set product
Binary relation
Simple map
Simple function
Measurable simple complex function
Simple integral
Unsigned basic integral
P-integrable basic function
Set of P-integrable complex Borel functions
Lebesgue length function
Definition D5594
Lebesgue distance function
Formulation 0
Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that
(i) $\mathfrak{L}^p = \mathfrak{L}^p (M \to \mathbb{C})$ is a D316: Set of P-integrable complex Borel functions on $M$
The Lebesgue distance function on $\mathfrak{L}^p$ is the D4367: Unsigned real function \begin{equation} \mathfrak{L}^p \times \mathfrak{L}^p \to [0, \infty), \quad (f, g) \mapsto \left( \int_X |f - g|^p \, d \mu \right)^{1 / p} \end{equation}