Definition:Space of Integrable Functions

Definition
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Then the space of $\mu$-integrable, real-valued functions $\mathcal{L}^1 \left({\mu}\right)$ is the collection of all $\mu$-integrable real-valued functions:


 * $\mathcal{L}^1 \left({\mu}\right) := \left\{{f: X \to \R: \text{$f$ is $\mu$-integrable}}\right\}$

Similarly, the space of $\mu$-integrable, extended real-valued functions $\mathcal{L}^1_{\overline{\R}} \left({\mu}\right)$ is the collection of all $\mu$-integrable extended real-valued functions:


 * $\mathcal{L}^1_{\overline{\R}} \left({\mu}\right) := \left\{{f: X \to \overline{\R}: \text{$f$ is $\mu$-integrable}}\right\}$

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 $\mathcal{L}^1 \left({\mu}\right)$ and $\mathcal{L}^1_{\overline{\R}} \left({\mu}\right)$ the space of $\mu$-integrable functions.

Furthermore, if necessary or convenient, it is common to write for example $\mathcal{L}^1 \left({\Sigma, \mu}\right)$ to emphasize $\Sigma$.

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

Also see

 * Integrable Function
 * Lebesgue Space, of which this is a special case