Category:Definitions/Lebesgue Spaces

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

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

The (real) Lebesgue $p$-space of $\mu$ is defined as:

$\map {\mathcal L^p} \mu := \set {f: X \to \R: f \in \map {\mathcal M} \Sigma, \displaystyle \int \size f^p \rd \mu < \infty}$

where $\map {\mathcal M} \Sigma$ denotes the space of $\Sigma$-measurable functions.

On $\map {\mathcal L^p} \mu$, we can introduce the $p$-seminorm $\norm {\, \cdot \,}_p$ by:

$\forall f \in \mathcal L^p: \norm f_p := \paren {\displaystyle \int \size f^p \rd \mu}^{1 / p}$

Next, define the equivalence $\sim$ by:

$f \sim g \iff \norm {f - g}_p = 0$

The resulting quotient space:

$\map {L^p} \mu := \map {\mathcal L^p} \mu / \sim$

is also called (real) Lebesgue $p$-space.