Category:Definitions/Measurable Functions
This category contains definitions related to Measurable Functions.
Related results can be found in Category:Measurable Functions.
Real-Valued Function
Definition 1
Let $\struct {X, \Sigma}$ be a measurable space.
Let $E \in \Sigma$.
Then a function $f: E \to \R$ is said to be $\Sigma$-measurable on $E$ if and only if:
- $\forall \alpha \in \R: \set {x \in E: \map f x \le \alpha} \in \Sigma$
Definition 2
Let $\struct {X, \Sigma}$ be a measurable space.
Let $E \in \Sigma$.
Let $\Sigma_E$ be the trace $\sigma$-algebra of $E$ in $\Sigma$.
Let $\map \BB \R$ be the Borel $\sigma$-algebra on $\R$.
Let $f : E \to \R$ be a real-valued function.
We say that $f$ is ($\Sigma$-)measurable if and only if:
Definition 3
Let $\struct {X, \Sigma}$ be a measurable space.
Let $E \in \Sigma$.
Then a function $f: E \to \R$ is said to be $\Sigma$-measurable on $E$ if and only if one of the following holds:
- $(1) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x \le \alpha} \in \Sigma$
- $(2) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x < \alpha} \in \Sigma$
- $(3) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x \ge \alpha} \in \Sigma$
- $(4) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x > \alpha} \in \Sigma$
Extended Real-Valued Function
Definition 1
Let $\struct {X, \Sigma}$ be a measurable space.
Let $E \in \Sigma$.
Then a function $f: E \to \overline \R$ is said to be $\Sigma$-measurable on $E$ if and only if:
- $\forall \alpha \in \R: \set {x \in E: \map f x \le \alpha} \in \Sigma$
Definition 2
Let $\struct {X, \Sigma}$ be a measurable space.
Let $E \in \Sigma$.
Let $\Sigma_E$ be the trace $\sigma$-algebra of $E$ in $\Sigma$.
Let $\map \BB {\overline \R}$ be the Borel $\sigma$-algebra on the extended real number space.
Let $f : E \to \overline \R$ be an extended real-valued function.
We say that $f$ is ($\Sigma$-)measurable if and only if:
Definition 3
Let $\struct {X, \Sigma}$ be a measurable space.
Let $E \in \Sigma$.
Then a function $f: E \to \overline \R$ is said to be $\Sigma$-measurable on $E$ if and only if one of the following holds:
- $(1) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x \le \alpha} \in \Sigma$
- $(2) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x < \alpha} \in \Sigma$
- $(3) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x \ge \alpha} \in \Sigma$
- $(4) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x > \alpha} \in \Sigma$
Subcategories
This category has the following 2 subcategories, out of 2 total.
Pages in category "Definitions/Measurable Functions"
The following 12 pages are in this category, out of 12 total.
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- Definition:Measurable Function
- Definition:Measurable Function/Banach Space Valued Function
- Definition:Measurable Function/Extended Real-Valued Function
- Definition:Measurable Function/Extended Real-Valued Function/Definition 1
- Definition:Measurable Function/Extended Real-Valued Function/Definition 2
- Definition:Measurable Function/Extended Real-Valued Function/Definition 3
- Definition:Measurable Function/Positive
- Definition:Measurable Function/Real-Valued Function
- Definition:Measurable Function/Real-Valued Function/Definition 1
- Definition:Measurable Function/Real-Valued Function/Definition 2
- Definition:Measurable Function/Real-Valued Function/Definition 3