Category:Definitions/Space of Measurable Functions Identified by A.E. Equality
This category contains definitions related to Space of Measurable Functions Identified by A.E. Equality.
Related results can be found in Category: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.
Pages in category "Definitions/Space of Measurable Functions Identified by A.E. Equality"
The following 2 pages are in this category, out of 2 total.