Measurable Function

Definition

$f:\Omega \to S\text{s.t.}\forall B\in \mathcal{S},f^{-1}(B)=\{\omega \in \Omega |f(\omega )\in B\}\in \mathcal{F}$

A measurable function is a function $f:\Omega \to S$ whose inverse function $X^{-1}(B)$ is $\mathcal{F}$-measurable the two measurable spaces $(\Omega ,\mathcal{F}),(S,\mathcal{S})$.

Other Expressions

$f:\Omega \to S$ is a measurable function on the two measurable spaces $(\Omega ,\mathcal{F}),(S,\mathcal{S})$ $f:(\Omega ,\mathcal{F})\to (S,\mathcal{S})$ $f$ is $\mathcal{F}-\mathcal{S}$ measurable $f\in \mathcal{F}$ (only if $(S,\mathcal{S})=(\mathbb{R},\mathcal{R})$) $f$ is a $S$-valued measurable

Examples

When $(S,\mathcal{S})=(\mathbb{R},\mathcal{R})$, the measurable function is called a Random Variable. When $(S,\mathcal{S})=({\mathbb{R}}^d,{\mathcal{R}}^d)$, the measurable function is called a Random Vector.

Facts

For the $\mathcal{F}$-measurable set $A\in \mathcal{F}$ on the two measurable spaces $(\Omega ,\mathcal{F}),(\mathbb{R},\mathcal{R})$ A function $f:\Omega \to \mathbb{R}$ defined as $f(\omega )=\{\begin{array}{ll}1& \omega \in A\\ 0& \omega \notin A\end{array}$ is a $\mathcal{F}-\mathcal{R}$ measurable function

If $f$ is a $\mathbb{R}$-valued measurable function on the two measurable spaces $(\Omega ,\mathcal{F}),(\mathbb{R},\mathcal{R})$, then $\forall \alpha \in \mathbb{R},\alpha f(\omega )$ also $\mathbb{R}$-valued measurable. Scaling of function does not change the Domain, only scale the Image

If $f:(\Omega ,\mathcal{F})\to (S,\mathcal{S})$ and $g:(S,\mathcal{S})\to (T,\mathcal{T})$ on the three measurable spaces $(\Omega ,\mathcal{F}),(S,\mathcal{S}),(T,\mathcal{T})$, then $g\circ f:(\Omega ,\mathcal{F})\to (T,\mathcal{T})$

If a function $f:\mathbb{R}\to \mathbb{R}$ is continuous, is measurable.

If two functions $f,g$ are $\mathbb{R}$-valued measurable on the two measurable spaces $(\Omega ,\mathcal{F}),(\mathbb{R},\mathcal{R})$, then following are all $\mathbb{R}$-valued measurable

• $h:=f+g$
• $h:=f-g$
• $h:=\max (f,g)$, where $h:\omega \mapsto \max (f(\omega ),g(\omega ))$
• $h:=\min (f,g)$, where $h:\omega \mapsto \min (f(\omega ),g(\omega ))$

If every element of a function sequence $\{f_n:n\in \mathbb{N}\}$ is $\mathbb{R}$-valued measurable, then the following are all $\mathbb{R}$-valued measurable.

• $h:=\inf f_n$, where $h:\omega \mapsto \inf \{f_1(\omega ),f_2(\omega ),\dots \}$
• $h:=\sup f_n$, where $h:\omega \mapsto \sup \{f_1(\omega ),f_2(\omega ),\dots \}$