Random Variable being $F$-measurable It is said the Random variable is $F$-measurable if $\{X\leq x\}$ is an element of $F$. Is $X$ not $F$measurable once it is not less than or equal to $1$ $x$ or only for all?
 A: Do you know what a $\sigma$-algebra is?  It is a collection of subsets of the sample space $\Omega$.  In particular, it is a collection of subsets that a) contains $\Omega$ as an element, b) is closed under set complement, and c) is closed under countable unions.
The purpose of a $\sigma$-algebra is to act as the domain of a measure $P$.  So the measure (often called a probability measure) $P$ is a function from our $\sigma$-algebra $\mathcal{F}$ to $[0,1]$.  When you input an element of $\mathcal{F}$ (which is a subset of $\Omega$), the measure outputs a number that we interpret as how likely that "event" is to occur.  The elements of $\mathcal{F}$ (which are subsets of $\Omega$) are called events.
Now, we say a function $X : \Omega \to \Bbb R$ is measurable if for each $\alpha \in \Bbb R$, $\{\omega \mid X(\omega) \leq \alpha \}$ (which is another way of writing $X^{-1}(-\infty, \alpha]$) is an event, i.e., if it is in $\mathcal{F}$, i.e., if we are able to assign a probability to it by evaluating it under the $P$ function/probability measure.
The reason we say $X$ is measurable under this definition is because it allows us to integrate the function $X$ using the Lebesgue integral.  Have you taken any analysis courses that introduce integration using measure theory?  If not, I think once you do, that will help you understand these definitions much more.  I actually think you should understand that material first before attempting to understand probability theory using measure theory, but that's just my opinion.
A: A random varible is defined on a probability space which consists of a sample space $\Omega$, a $\sigma$-algebra $\mathcal{F}$ on $\Omega$ and a measure $P:\mathcal{F}\rightarrow [0,1]$.  
In this case, $X$ is a random variable if for ever $x\in\mathbb{R}$, 
$[X\le x]\in \mathcal{F}$.  To wit, $X$ is a measurable function on the measure space $(\Omega, \mathcal{F})$.
