Clarifying the importance of the quantile function in probability theory I want to cement my understanding of the quantile function in probability theory and here is the way I understand it.
(1) We start off with some probability space $(\mathbb R, B = \sigma(\mathbb R),\mathbb P)$.
(2)Define the CDF of $\mathbb P$ as $F_{\mathbb P}=\mathbb P((-\infty,c])~\forall c \in \mathbb R$
(3) Let $X(u) = inf\{c \in \mathbb R : F_{\mathbb P}(c) \ge u    \} ~~\forall u \in (0,1)$
Then, by the following lemma: Every probability meausure on $(\mathbb R, B)$ is already uniquely determined by some CDF function, X is a random variable
(4) $X:(0,1) \to \mathbb R$
(5) New Probability space $( (0,1), B (0,1)=\sigma((0,1)), \mathscr U_{(0,1)})$ where $\mathscr U$ is the uniform probability measure.
(6)By definition of a random variable, $A \ \in B(\mathbb R)~\implies X^{-1}(A) \in B((0,1) )$.
I hope this logic makes sense, now the importance of this is, as I understand it, is that for any probability measusre, we are able to find a random variable from the uniform probability space back to the original space. And this is very useful, for example in monte carlo simulation if we are trying to simulate from some distribution, and we are not sure how, we can simulate a uniform and plug it into the quantile?
Any intuition is appreciated!
 A: Most random number generators return pseudorandom observations from $Unif(0,1)$. The high-quality generators used in statistical software such as R are vetted by huge batteries of tests to confirm that they behave, for all known practical purposes, as if they were independent observations from $Unif(0,1).$ 
However, most simulation (or Monte Carlo) work involves distributions other than uniform. If the cumulative distribution function (CDF) of a distribution can be expressed in closed form and can be inverted to get the quantile function, then one method of simulation from that distribution is to use the quantile function as you have said. Even in these circumstances, the quantile function may not be best. [For example, if the quantile function has an extensive region that is essentially flat, even double precision uniform observations may be too widely separated to give a realistic variety of values upon transformation. Digital computers do not deal with the real numbers, but with a convenient--and very frequently adequate--subset of the rationals.]
A simple example of the quantile method at its best is to generate observations from
the distribution $BETA(0.5, 1),$ which has density function $f(x) = 0.5x^{-0.5}$ on $(0, 1)$, and 0 elsewhere. Its CDF is $F(x)= \sqrt{x}$ on
$(0, 1)$, (and 0 below, 1 above). Thus, upon taking the inverse of the CDF, the quantile transform method is to square observations from $Unif(0,1)$ to obtain observations from $BETA(0.5,1).$ Some intuition can be obtained by observing that a tenth of the observations from the uniform distribution come from $(0,.1),$ but upon squaring endpoints, one sees that transformed values are crowded into the interval $(0, .01)$. Similarly, upon squaring, the one tenth of values originating in $(.9, 1)$ get stretched out over $(.81, 1)$ upon transformation. 
There is rich literature on algorithms for turning uniform variates into observations from other distributions by methods other than quantile transformation.
Because the CDF of a standard normal distribution cannot be expressed in closed form, it is not computationally efficient to use the quantile method to simulate a sample from the normal family. You can google 'Box-Muller generator' for presentations of the most common method of doing so. (Ultimately the Box-Muller method uses two uniform pseudorandom variates to generate two standard uniform ones.)
However, one must realize that all of these 'non-quantile' methods also use (pseudo)random samples uniformly distributed on $(0, 1)$ to generate whatever probability structure may be at hand. Sometimes millions of uniform variates are used in a modern
simulation. But ultimately everything comes from inside a multi-dimensional unit cube.
Please feel free to post a comment if elaboration of any of this might be helpful.
