The definition I have in my notes is as follows:
Given a $\sigma$-algebra $\mathcal{G} \subseteq \mathcal{F}$, we say that $X$ is $\mathcal{G}$-measurable or measurable with respect to the $\sigma$-algebra $\mathcal{G}$ if $\{X \leq x\} \in \mathcal{G}$ for all $x \in \mathbb{R}$.
I don't really understand what to look for to check whether $\{X \leq x\}$ is in $\mathcal{G}$.
There is one half example in my notes which is as follows:
An integrable random variable $X$, define the random variable $Z$ by $$ Z(\omega)=\left\{\begin{array}{ll} \frac{\mathbb{E}\left(X I_{B}\right)}{P(B)} & \text { if } \omega \in B \\ \frac{\mathbb{E}\left(X I_{B}\right)}{P\left(B^{c}\right)} & \text { if } \omega \in B^{c} \end{array}\right. $$ Then $Z$ is $\mathcal{G}$-measurable (why?)
Which as you can see, frustratingly leaves the task of showing $\mathcal{G}$-measurability to the reader. I'm really not sure how I can apply the above definition to this. If anyone could show that some function is $\mathcal{G}$-measurable with either this example or their own, or a brief explanation of what to look for to show this, it would be greatly appreciated. I think a worked example will help me understand what I need to look for and how I can apply a similar method to other functions or just an explanation that