A asked a related question before: Exercise 2.2 on measuriability criterion from Probability for Statisticians . I didn't have much idea of the concepts back then but had a slightly better understanding of the topic now. I modified my answer according to what Jakobian has suggested. However, I am still very confused with what this excercise is trying to convey. This post is related to general understanding of the question. The question is as following:
Let $\mathcal{C}$ denote a $\bar{\pi}$ -system of subsets of $\Omega .$ Let $\mathcal{V}$ denote a vector space of functions; that is, $X+Y \in \mathcal{V}$ and $\alpha X \in \mathcal{V}$ for all $X, Y \in \mathcal{V}$ and all $\alpha \in R$ and, all the usual elementary facts hold.
- Suppose that: $ 1_{C} \in \mathcal{V} $ for all $C \in \mathcal{C}$. If $A_{n} \nearrow A$ with $1_{A_{n}} \in \mathcal{V}, $ then $1_{A} \in \mathcal{V}$ Show that $1_{A} \in \mathcal{V}$ for every $A \in \sigma[\mathcal{C}]$.
- It then follows trivially that every simple function $X_{n} \equiv \sum_{1}^{m} \alpha_{i} 1_{A_{i}} $ is in $\mathcal{V}$ here $m \geq 1,$ all $\alpha_{i} \in R,$ and $\sum_{1}^{m} A_{i}=\Omega$ with all $A_{i} \in \sigma[\mathcal{C}]$
- Now suppose further that $X_{n} \nearrow X$ for $X_{n}$ 's as in 2. implies that $X \in \mathcal{V} .$ Show that $\mathcal{V}$ contains all $\sigma[\mathcal{C}]$ -measurable functions.
To my understanding this question is a sort of application to the measurability criteria and the form of all $\mathcal{F}(Z)$-measurable function that has been stated earlier in the textbook:
Proposition (Measurability criteria) $\quad$ Let $X: \Omega \rightarrow \bar{R} .$ Suppose $\sigma[\mathcal{C}]=\overline{\mathcal{B}}$ Then measurability can be characterized by either of the following:
- $X$ is measurable if and only if $X^{-1}(\mathcal{C}) \subset \mathcal{A}$
- $X$ is measurable if and only if $X^{-1}([-\infty, x]) \in \mathcal{A}$ for all $x \in \bar{R}$.
Proposition (The form of an $\mathcal{F}(\mathbf{Z})$ -measurable function) Suppose that $Z$ is a measurable function on $(\Omega, \mathcal{A})$ and that $Y$ is $\mathcal{F}(Z)$ -measurable. Then there must exist a measurable function $g$ on $(\bar{R}, \overline{\mathcal{B}})$ such that $Y=g(Z)$.
It seems to me the questino is a step-by-step guidance for us to find forms of $\sigma[\mathcal{C}]$-measurable function defined on a vector space?
But in this case $\sigma[\mathcal{C}]$ is not a $\overline{\mathcal{B}}$ where $\mathcal{C}$ in this case is not any of $\mathcal{C}_I\equiv \{(a,b],(-\infty,b],\text{or},(a,\infty):-\infty<a<b<\infty\}$, $\mathcal{C_F}\equiv\{\text{all finite disjoint unions of intervals in } \mathcal{C}_I\}$. In my understanding if we have $X:(\Omega,\mathcal{A})\rightarrow (\overline{R},\overline{\mathcal{B}})$, $X$ will be a measurable function. I am very confused what should be the corresponding $X$ in this question? I am sorry, I am very confused with this question and I hope my question make sense. If not, any please points out the non-sense part of my question so that I can make my question more specific and make my mind more clear.