Note: I asked this on Maths Ed SE, but it was deleted automatically.
So, I recently (re-)discovered that random variables learned in elementary probability such as the exponentially distributed random variable $X$ with cdf $F_X(x) = 1-e^{- \lambda x}$ can be explicitly represented as
$$X(\omega) := \frac{1}{\lambda} \ln(\frac{1}{1-\omega})$$
where the probability space is $((0,1), \mathscr B(0,1), \mu)$ where $\mu$ is Lebesgue measure.
This can be derived with the formula
$$X(\omega) = \sup\{x \in \mathbb{R}: F_X(x) < \omega\}$$
This is apparently called Skorokhod representation (so-called in David Williams' Probability with Martingales).
Now, I guess you don't need measure theory to understand quantile functions and do NORMINV(RAND(),\mu,\sigma). You just say that $F_X^{-1}(Y)$ has the same distribution as $X$. Well, the $F_X^{-1}$ may need some measure theory, but I'm more interested in proving that $X(\omega) = \frac{1}{\lambda} \ln(\frac{1}{1-\omega})$ is exponential with parameter $\lambda$. I was thinking:
$$P(\frac{1}{\lambda} \ln(\frac{1}{1-\omega}) \le x)$$
$$ = P(\omega \le 1-e^{-\lambda x})$$
$$ = F_U(1-e^{-\lambda x})$$
where $U \sim Unif(0,1)$.
I guess we have to take without explanation that the Skorokhod representation of uniform distribution is $U(\omega) = \omega$, but once we do then we don't need Lebesgue measure, probability spaces, measure spaces, etc to say that $P(\omega \le 1-e^{-\lambda x}) = 1-e^{-\lambda x}$