Category:Definitions/Lp Spaces

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This category contains definitions related to Lp Spaces.
Related results can be found in Category:Lp Spaces.


Let $\struct {X, \Sigma, \mu}$ be a measure space, and let $p \in \closedint 1 \infty$.

Let $\map \MM {X, \Sigma, \R}$ be the set of real-valued $\Sigma$-measurable functions on $X$.

Let $\map {\LL^p} {X, \Sigma, \mu}$ be the Lebesgue $p$-space of $\struct {X, \Sigma, \mu}$.

Let $\sim_\mu$ be the restriction of the $\mu$-almost-everywhere equality relation on $\map \MM {X, \Sigma, \R}$ to $\map {\LL^p} {X, \Sigma, \mu}$.


We define the $L^p$ space $\map {L^p} {X, \Sigma, \mu}$ as the quotient set:

\(\ds \map {L^p} {X, \Sigma, \mu}\) \(=\) \(\ds \map {\LL^p} {X, \Sigma, \mu} / \sim_\mu\)
\(\ds \) \(=\) \(\ds \set {\eqclass f {\sim_\mu}: f \in \map {\LL^p} {X, \Sigma, \mu} }\)
\(\ds \) \(=\) \(\ds \set {\set {g \in \map {\LL^p} {X, \Sigma, \mu} : f = g \, \mu\text{-almost everywhere} }: f \in \map {\LL^p} {X, \Sigma, \mu} }\)

Subcategories

This category has only the following subcategory.