Sigma Algebra Measurable R.V I am trying to figure out what random variables are measurable with respect to sigma algebra given by $[1-4^{-n}, 1]$ where $n= 0, 1, 2, ....$ if $[0,1]$ is the sample space. I believe I can do with with indicator functions but I'm not sure how to write this. 
Thanks!

If $\mathcal{P}$ is a countable partition of the space then $\mathcal{F}=\left\{ \cup\mathcal{P}'\mid\mathcal{P}'\subset\mathcal{P}\right\} $
is the $\sigma$-algebra generated by the elements of $\mathcal{P}$.
A function $f$ is $\mathcal{F}$-measurable iff it is constant on
elements of $\mathcal{P}$. 
So such functions take the form: $$\sum_{P\in\mathcal{P}}c_{P}1_{P}$$
where the $c_{P}$ are constants and the $1_{P}$ are characteristic
functions. 
You are in such a position. The sets $I_{n}=\left[1-4^{-n},1-4^{-n-1}\right)$
together with singleton $\left\{ 1\right\} $ form such a partition of
$\left[0,1\right]$.
