# Category:Space of Measurable Functions Identified by A.E. Equality

This category contains results about Space of Measurable Functions Identified by A.E. Equality.
Definitions specific to this category can be found in Definitions/Space of Measurable Functions Identified by A.E. Equality.

### Real-Valued Function

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

Let $\map \MM {X, \Sigma, \R}$ be the set of real-valued $\Sigma$-measurable functions on $X$.

Let $\sim_\mu$ be the almost-everywhere equality relation on $\map \MM {X, \Sigma, \R}$ with respect to $\mu$.

We define the space of real-valued measurable functions identified by $\mu$-A.E. equality as the quotient set:

 $\ds \map \MM {X, \Sigma, \R}/\sim_\mu$ $=$ $\ds \set {\eqclass f {\sim_\mu} : f \in \map \MM {X, \Sigma, \R} }$

### Extended Real-Valued Function

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

Let $\map \MM {X, \Sigma}$ be the set of $\Sigma$-measurable functions on $X$.

Let $\sim_\mu$ be the almost-everywhere equality relation on $\map \MM {X, \Sigma}$ with respect to $\mu$.

We define the space of real-valued measurable functions identified by $\mu$-A.E. equality as the quotient set:

 $\ds \map \MM {X, \Sigma, \R} / \sim_\mu$ $=$ $\ds \set {\eqclass f {\sim_\mu} : f \in \map \MM {X, \Sigma, \R} }$

## Subcategories

This category has only the following subcategory.