Definition(Indicator Function)

For a set \(A\subset\Omega\), the function \(I_A:\Omega\rightarrow\mathbb{R}\) defined by \[ I_A(\omega)=\begin{cases}1, & \mbox{if }\omega\in A,\\0,& \mbox{otherwise}.\end{cases} \] is called the indicator of \(A\).

  

Remark

Note that \(I_A\) is measurable-\(\mathcal{F}/\mathcal{R}\) if and only if \(A\in \mathcal{F}\) because

\[ I_A^{-1}(H)=\begin{cases}A, & \mbox{if }1 \in H,0 \notin H\\A^c,& \mbox{if }1 \notin H,0 \in H \\ \Omega & \mbox{if }0,1\in H\\ \phi, & \mbox{if }0,1\notin H,\end{cases} \] so that \(I_A^{-1}(H)\in \mathcal{F}\) for all \(H\in \mathcal{R}\) if and only if \(A\in \mathcal{F}\).

  

Definition(Simple function)

A real-valued simple function is a function \(f:\Omega\rightarrow\mathbb{R}\) that can be written in the form \[ f(\omega)= \sum_{i=1}^n a_iI_{A_i}(\omega), \] where \(a_1,\ldots,a_n\in \mathbb{R}\) and \(A_1,\ldots,A_n\) form a finite partition of \(\Omega\)(i.e., \(\Omega=\uplus_{i=1}^nA_i)\).

  

Theorem

If \(f:\Omega\rightarrow \mathbb{R}\) is nonnegative and \(\mathcal{F}\)-measurable, then there exists a sequence \(\{f_n\}\) of \(\mathcal{F}\)-measurable, real-valued simple functions such that \(0\le f_n(\omega)\uparrow f(\omega)\) for all \(\omega\in \Omega\).

  

Corollary

If \(f:\Omega \rightarrow \mathbb{R}\) is \(\mathcal{F}\)-measurable, then there exists a sequence \(\{f_n\}\) of \(\mathcal{F}\)-measurable, real-valued simple functions such that \[ 0\le f_n(\omega)\uparrow f(\omega) \mbox{ if } f(\omega)\ge 0,\\ \] and \[ 0\ge f_n(\omega)\downarrow f(\omega) \mbox{ if } f(\omega)\le 0.\\ \]

  

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