Defining the probability distribution function Suppose $X$ is a random variable that takes values in $[0,1].$ Is it that the probability distribution of $X$ is defined on $[0,1]$ or whole $R$?
The probability distribution is defined on the whole of $\mathbb{R}$. Also, you need to understand that, this is given by an integral or sum depending on what $X$ is, whose limits are usually taken to be from $-\infty$ to $x \in \mathbb R$.

In general, any function $F$ is a distribution function if it satisfies the following properties:


$0 \leq F(x) \leq 1$ for all $x \in \mathbb{R}$
$F$ is a non-decreasing function of $x$.
$F(-\infty) := \lim_{x \to -\infty}{F(x)}=0$ and $F(\infty) := \lim_{x \to \infty}{F(x)}=1$
$F$ is right continuous, i.e. $F(x+)=F(x)$ for all $x \in \mathbb R$.


And, it can be shown that, given $F$ with above properties, there exists a probability space $(\Omega, \mathscr A, P)$ on which a random variable $X$ exists  such that $X$ has $F$ as its distribution.
There is not a lot of difference. 

With a probability space $(\Omega, \mathscr A, P)$, the sample space $\Omega$ must contain all possible events; whether it includes impossible ones too does not matter much so long as you are clear.

If the distribution is defined on the whole of $\mathbb{R}$ but values cannot be less than 0 or more than 1, then you can ignore both of these possibilities.  If you keep them then $\Pr(X\lt 0)=0$ and $\Pr(X\gt 1)=0$.