Can some probability triple give rise to any probability distribution? Suppose we have a probability triple $(\Omega,\mathcal{F},P)$ and random variable $X:\Omega\to(\mathbb{R},\mathcal{B})$ with $\mathcal{B}$ denoting the Borel $\sigma$-algebra. Then, the distribution function of $X$ can be defined as
$$
F(x)=P(\{w\in\Omega:X(w)\leq x\})\tag{$*$}.
$$
Clearly, the RHS of ($*$) depends on $\Omega$, $P$, and the function $X$ whereas the LHS doesn't have to share such dependence. For example, we can define the LHS as the normal CDF $F(x)=\int_{-\infty}^x\frac{1}{\sqrt{2\pi}}\exp(-s^2/2)ds$. So I have two related questions:


*

*Given $(\Omega,\mathcal{F},P)$, how do we then know all the possible $F(\cdot)$'s that can be induced by appropriately defined random variables i.e. measurable functions from $(\Omega,\mathcal{F})$ to $(\mathbb{R},\mathcal{B})$?

*Is there some $(\Omega,\mathcal{F},P)$ such that for any CDF $F(\cdot)$, there is some random variable $X:(\Omega,\mathcal{F})\to(\mathbb{R},\mathcal{B})$ such that the distribution of $X$ is $F$?


If the questions are unclear, I'll be happy to edit / clarify. I would also appreciate it if you could include some references in your answers. Thanks!
 A: It matters what your definition is of a probability measure but here is a start and it addresses the situation where $\Omega$ is a subset of the reals.  For the first question.  There is an equivalence between CDFs and probability measures on the space.  As was suggested I will give that correspondance:
     $$\mu([-\infty, x]) = F(x)$$
Then $\mu$ can be fully extended to a measure via the Caratheodory Extension Theorem.
Given a probability measure $\mu$ you can appeal to the Lebesgue Decomposition Theorem to show that it has two components wrt to $P$.  The first is an absolutely continuous measure wrt to $P$.  The second is singular.  That means you can write the first in terms of its Radon-Nikodym derivative wrt $P$ and the integral of that function is part of the $F(x)$.  The issue though is that there can be a singular part.  To introduce "functions" that represent them you will need to expand your set to distributions to includes the likes of the dirac mass.  See the Radon Nikodym Theorem for more details.
For the second question.  You can define the measure via it's Radon-Nikodym derivative wrt to Lebesgue measure $P$ by setting $\frac{d\mu}{dP} = \frac{d}{dx} F(x)$ where you have to generalize your of derivative to allow the derivative at points where $F$ is discontinuous to result in dirac masses when you take the derivative (i.e. you need the Distributional derivative).  A simple example of that, say $F(x)$ is 0 for $x<0$ and $1$ for $x>=0$.  Then the Radon-Nikodym derivative is a dirac mass at $0$.
Essentially then for such an $\Omega$ the allowable functions $F$ are those that are measurable, non-decreasing, are $0$ off at $-\infty$ and $1$ at $\infty$.
A: The "universal probability space" in the sense of your second question is $([0,1],\mathcal{B}([0,1]),m)$ where $m$ is Lebesgue measure. The procedure for making the identification is the procedure that is commonly used for numerical sampling of a random variable given its CDF. Specifically, we uniformly choose $p \in [0,1]$, then choose the smallest $x$ such that $F(x) \geq p$. This mapping from $p$ to $x$ is a well defined function of $p$; I will call it $F^+$. (In general $F$ is not bijective, so I should not call it $F^{-1}$.) The result is a random variable, because $F^+$ is a nondecreasing function, hence Borel measurable.
