ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Binary cartesian set product
Binary relation
Map
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 D117
Complex Lebesgue quotient set
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 $M$
(ii) $\Vert \cdot \Vert_{\mathfrak{L}^p}$ is the D317: Lebesgue length function on $\mathfrak{L}^p$
(iii) \begin{equation} {\sim} : = \left\{ (f, g) \in \mathfrak{L}^p \times \mathfrak{L}^p : \Vert f - g \Vert_{\mathfrak{L}^p} = 0 \right\} \end{equation}
The complex Lebesgue quotient set on $M$ with respect to $p$ is the D180: Quotient set \begin{equation} \mathfrak{L}^p / {\sim} \end{equation}
Children
Complex random Lebesgue quotient set