Check/Prove if random variable I am having some doubts on how to prove/check if something is a random variable.
The question I am trying to solve currently is:
1) Consider X a random variable on space (Ω,F,P). If [X] is the biggest integer  equals or greater than X. Is [X] a random variable?
2) If (Ω,F,P) is a probability space and A1, A2... An spaces in F. Check if 
X = Σ($a_i$$\{1(A_i) \}$) (from i=1 to n) and 
Y = Π($1(A_i)$) (from 1 to n)  are random variables.
I was trying to see if I could find some kind of properties that random variables some have, but all I could find were properties that a distribution function F of a random variable should have. These were:
i) 0≤F(x)≤1,x∈.
ii) F is nondecreasing.
iii) F is continuous from the right.
iv) F(x) → 0 as x → −∞, F(x) → 1, as x → +∞.
We express this by writing F(−∞) = 0, F(+∞) = 1.
Now, if I prove that my random variable above have an F that satisfies those conditions, do I also prove that they are random variables? If this isn't true, how should I proceed? Is there a way with measure theory that I'm not aware?
Sorry if I made anything confusing, my questions were not in english so I had to do a bit of translating.
 A: You are on the wrong track. To be checked is actually whether the functions are measurable functions $:\Omega\rightarrow\mathbb R$.
On 1): 
Let it be that $x=n+r$ where $n$ is an integer and $r\in\left[0,1\right)$.
Preassuming that $\left[X\right]$ is the smallest integer that is
not exceeded by $X$ we find that $\left\{ \left[X\right]\leq x\right\} =\left\{ X\leq n\right\} $
and the fact that $X$ is a random variable tells us that this is
a measurable set. This is true for every $x\in\mathbb{R}$ so we are
allowed to conclude that $\left[X\right]$ is a random variable.
On 2): 
The sets that can be written as $E_{1}\cap\cdots\cap E_{n}$ where
$E_{i}\in\left\{ A_{i},A_{i}^{c}\right\} $ are all measurable, and
so are unions of these sets. If $X:=\sum_{i=1}^{n}a_{i}1_{A_{i}}$
then $\left\{ X\left(\omega\right)\mid\omega\in\Omega\right\} $ is
a finite set and for every $c\in\mathbb{R}$ the set $\left\{ X=c\right\} $
is an (eventually empty) union of these sets. This tells us that $X$
is a measurable function.
Note that $Y$ is the characteristic set of intersection $\bigcap_{i=1}^{n}A_i$.
