Problem:
I have a good understanding of basic Bernoulli and Binomial RVs, but this was foundational work in statistics. I am attempting to try and apply my (minimal but increasing) knowledge of measure theory to a tangible concept. I have been working with simple functions, etc. and am trying to utilize only these tools to find expectation:
if $f=\sum_{i=1}^m c_i1_{A_i}$ has distinct, finite c's and disjoint A's, then $\int f du=\sum_{i=1}^m c_i\mu(A_i)$ and if $f$ is measurable and $f_n \uparrow f$ then $\int f du=\lim_{n\rightarrow\infty}\int f_n d\mu$
I want to try and practice (read: learn how to) utilize these ideas on a measure space of an infinite number of Bernoulli trials. I define my space below:
Work
Take $(\Omega,\mathcal{B})=(\{0,1\}^{\infty},\mathcal{B}(\{0,1\}^{\infty}))$ and define an event $\omega\in\Omega$ as $\omega=(x_1,x_2,...)$
Then I defined a probability measure: $P(\{x_1\}$x{0,1}x...$)=\prod_{i=1}^n p^{x_i}(1-p)^{1-x_i}$.
From here I want to find the expectation of RVs such as:
1- $Z(\omega)=\sum_{i=1}^nx_i$
2- $Y=e^{sZ}$ (moment-generating function), and
3- $V(\omega)=\sum_{i=1}^{\infty}r^nx_n$ for positive r.
Using comments below:
$Z(\omega)=\sum_{i=1}^n 1_{A_i}, A_i\subset\Omega, A_i=\{(x_k)_{k\ge 1}\in\Omega|x_i=1\}$
$Y_n(x_1,...,x_n)=exp \left(s\sum_{i=1}^n x_i \right )$
$E_n(Y_n)=\sum_{(x_k)\in\Omega_n}exp \left(s\sum_{i=1}^n x_i \right )\prod_{i=1}^n p^{x_i}(1-p)^{1-x_i}$
$E(Y)=(1-p+pe^s)^n$
$E(V)=\sum_{i=1}^n r^i p, \forall n$
$V(\omega)_n \uparrow V(\omega)\rightarrow E(V(\omega)_n)\uparrow E(V(\omega))$
I showed these to a friend and he had the following comments:
1-Each RV needs to be represented as a simple function, a limit of a nondecreasing sequence of non-negative functions, or a difference of two such limits (whose product is zero).
For example, on $Z$, you need to compute $E(Z)=\sum_{i=1}^n P(A_i)$ and show how each $P(A_i)$ is derived from $P$ as being a unique probability measure satisfying the equality.
Also, if the RV is a limit of simple functions, you have to find the expectation of the simple function in the sequence and take the limit.
Given that I am learning this on my own from scratch, any explicit help would be greatly appreciated. No detail is too much!