Category:Space of Real-Valued Measurable Functions Identified by A.E. Equality
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This category contains results about Space of Real-Valued Measurable Functions Identified by A.E. Equality.
Definitions specific to this category can be found in Definitions/Space of Real-Valued Measurable Functions Identified by A.E. Equality.
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:
| \(\ds \map \MM {X, \Sigma, \R}/\sim_\mu\) | \(=\) | \(\ds \set {\eqclass f {\sim_\mu} : f \in \map \MM {X, \Sigma, \R} }\) |
Pages in category "Space of Real-Valued Measurable Functions Identified by A.E. Equality"
The following 5 pages are in this category, out of 5 total.
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- Pointwise Addition on Space of Real-Valued Measurable Functions Identified by A.E. Equality is Well-Defined
- Pointwise Multiplication on Space of Real-Valued Measurable Functions Identified by A.E. Equality is Well-Defined
- Pointwise Scalar Multiplication on Space of Real-Valued Measurable Functions Identified by A.E. Equality is Well-Defined