Definition:Integral on L-1 Space

Definition

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

Let $\map {\mathcal L^1} {X, \Sigma, \mu}$ be the Lebesgue $1$-space of $\struct {X, \Sigma, \mu}$.

Let $\map {L^1} {X, \Sigma, \mu}$ be the $L^1$ space of $\struct {X, \Sigma, \mu}$.

We define the integral $I_\mu : \map {L^1} {X, \Sigma, \mu} \to \R$ by:

$\ds \map {I_\mu} {\eqclass f \sim} = \int f \rd \mu$

where:

$\eqclass f \sim \in \map {L^1} {X, \Sigma, \mu}$, where $\eqclass f \sim$ is the equivalence class of $f \in \map {\LL^1} {X, \Sigma, \mu}$ under the $\mu$-almost everywhere equality relation.

$\ds \int \cdot \rd \mu$ denotes the usual $\mu$-integral of a $\mu$-integrable function

More crudely we write:

$\ds \int \eqclass f \sim \rd \mu = \int f \rd \mu$

Also see

• Integral on $L^1$ Space is Well-Defined