I am seeking to establish a rigorous operationalization of “choosing an element randomly” from an arbitrary abstract sample space. The idea is as follows. First, choose a number between $0$ and $1$ according to the uniform distribution. (This can be modeled in quite a straightforward way through an infinite sequence of tosses of a fair coin.) Then, the number between $0$ and $1$ chosen determines the element you choose from whatever abstract sample space in such a way that conforms to the probability measure on the abstract space.
Formally, let $(\Omega,\mathscr F,\mathbb P)$ be an arbitrary probability space. Moreover, let $((0,1],\mathscr B,\mu)$ be the unit interval endowed with the Borel $\sigma$-algebra $\mathscr B$ and the Lebesgue measure $\mu$. The problem is to construct a function $T:(0,1]\to\Omega$ (mapping the outcome of the choice of the number into selection on the sample space) that is $\mathscr B/\mathscr F$-measurable and such that $$\mu\left(x\in(0,1]\,|\,T(x)\in A\right)=\mathbb P(A)\quad\text{for each $A\in\mathscr F$}.$$ Note that the probability measure $\mathbb P$ is the pushforward measure generated by the Lebesgue measure via the map $T$: $\mathbb P(A)=\mu(T^{-1}(A))$ for each $A\in\mathscr F$.
My question is whether and how to construct such a sampling function $T$ and whether doing so is always possible in the first place for an arbitrary given abstract probability space $(\Omega,\mathscr F,\mathbb P)$. Any feedback would greatly appreciated.
Counterexample:
Let $\Omega$ be a set of cardinality $> \frak c$, $\mathscr F$ the $\sigma$-algebra of subsets $S$ of $\Omega$ such that either $S$ or its complement has cardinality $\le \frak c$, $\mathbb P(S) = 0$ if $\text{card}\; S \le \frak c$, $1$ otherwise. In particular, if $A = T((0,1])^c$ we have $\mathbb P(A)= 1$ but $\mu(T^{-1}(A)) = \mu(\emptyset)=0$.