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!
 A: The collection of intervals $[1-4^{-n}]$, $n\geqslant 0$ does not form a $\sigma$-algebra, but if we define $I_n:=[1-4^{-n},1-4^{-n-1})$, then the collection of sets of the form $\bigcup_{n\in J}I_n$, $J\subset\mathbb N$, plus singleton $\{1\}$,  is a $\sigma$-algebra. 
If a $\sigma$-algebra is generated by a countable partition $(A_n)_{n\in\mathbb N}$, we can find an explicit characterization of measurable functions, namely, functions of the form $x\mapsto \sum_{n=0}^{+\infty}c_n\chi_{A_n}$, where $c_n$'s are real numbers. 
A: Well, any indicator function won't do. You can see that there is no way to write $(0,25,0.5)$ as an element of the $\sigma$-algebra generated by those sets, and so the indicator of that set won't be measurable. But the set of simple functions which are measurable in that space is most likely dense with respect to any measure that is absolutely continuous with respect Lebesgue measure.
A: 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
$\left[0,1\right]$.
