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 $\struct {X, \Sigma, \mu}$.

Let $f \in \map {\mathcal L^1} {X, \Sigma, \mu}$.

Let $\sim$ be the almost-everywhere equality equivalence relation on the set of $\Sigma$-measurable functions $\map {\mathcal M} {X, \Sigma}$.

Let $\eqclass f \sim \in \map {L^1} {X, \Sigma, \mu}$ be the equivalence class of $f$ under $\sim$.

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 $\ds \int \cdot \rd \mu$ 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