Definition:Space of Integrable Functions

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Let $\struct {X, \Sigma, \mu}$ be a measure space.

Then the space of $\mu$-integrable, real-valued functions $\map {\LL^1} \mu$ is the collection of all $\mu$-integrable real-valued functions:

$\map {\LL^1} \mu := \set {f: X \to \R: \text {$f$ is $\mu$-integrable} }$

Similarly, the space of $\mu$-integrable, extended real-valued functions $\map {\LL^1_{\overline \R} } \mu$ is the collection of all $\mu$-integrable extended real-valued functions:

$\map {\LL^1_{\overline \R} } \mu := \set {f: X \to \overline \R: \text {$f$ is $\mu$-integrable} }$

Also known as

It is often taken clear from the context whether the functions are real-valued or extended real-valued.

Thus, one often simply calls $\map {\LL^1} \mu$ and $\map {\LL^1_{\overline \R} } \mu$ the space of $\mu$-integrable functions.

Furthermore, if necessary or convenient, it is common to write for example $\map {\LL^1} {\Sigma, \mu}$ to emphasize $\Sigma$.

When $\mu$ is clear from the context, it may also be dropped from both name and notation, yielding $\LL^1$, the space of integrable functions.

Also see