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:

$f$ is $\Sigma_E \, / \, \map \BB \R$-measurable.


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:

$(1) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x \le \alpha} \in \Sigma$
$(2) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x < \alpha} \in \Sigma$
$(3) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x \ge \alpha} \in \Sigma$
$(4) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x > \alpha} \in \Sigma$


Also see