Random variable defined on the Lebesgue probability space

There is a random variable X defined on the Lebesgue probability space whose cumulative distribution function is F. We can find X(w) knowing that:

$$X(ω)=\inf\{x∈R:F(x)>ω\}$$.

1) how do we prove that for the uniform random variable (in the Lebesgue probability space) on $$[a,b]$$, :

$$X(ω)=a + (b-a)ω$$

2) for a discrete random variable (f.i the Bernoulli distribution) how does this formula applies ?

3) A similar problem: how to prove that Y(w) = 1 - w is a random variable, and how to give its distribution? My guess is that we use $$P(Y \le x) := P(\omega | Y(\omega) \le x)$$

$$Y(w)\le x$$ $$\iff w\ge 1-x$$ Then we have : $$P(\omega | Y(\omega) \le x)= (1-x) 1 _{x\ge0}\$$ Am i on the right path?