# Definition:Measurable Function

## 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:

$f$ 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:

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

## 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:

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

### 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:

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

## Positive Measurable Function

Let $\struct {X, \Sigma}$ be a measurable space.

Let $S \in \set {\R, \overline \R}$.

Let $f : X \to S$ be a $\Sigma$-measurable function.

We say that $f$ is a positive $\Sigma$-measurable function if and only if:

$f \ge 0$

where $\ge$ denotes pointwise inequality.

## Banach Space Valued Function

Let $\GF \in \set {\R, \C}$.

Let $I$ be a real interval.

Let $X$ be a Banach space over $\GF$.

Let $f : I \to X$ be a function.

We say that $f$ is measurable if there exists a sequence of simple functions $\sequence {f_n}_{n \mathop \in \N}$ such that:

$\ds \map f t = \lim_{n \mathop \to \infty} \map {f_n} t$

for Lebesgue almost all $t \in I$.