Definition:Measurable Function/Real-Valued Function
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Definition
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:
- $(1) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x \le \alpha} \in \Sigma$
- $(2) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x < \alpha} \in \Sigma$
- $(3) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x \ge \alpha} \in \Sigma$
- $(4) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x > \alpha} \in \Sigma$