examples of random variables that are not measurable Let $(\Omega,\mathcal{F},\mathbb{P})$ a probability space and $(\Omega',\mathcal{F}')$ a measurable space. Furthermore let
\begin{equation}
X:(\Omega,\mathcal{F},\mathbb{P}) \rightarrow (\Omega',\mathcal{F}')
\end{equation}
be any function. Can someone give examples where $X$ is not measurable?
Small note: we only call such a function a random variable when $X$ is measurable.

A very simple example would be $\Omega = \Omega' = \{0, 1\}$, with $\mathcal F = \{\emptyset, \Omega\}$ and $\mathcal F' = \mathcal P(\Omega)$, and $X = \mathrm{id}_\Omega$. Then $\{1\}$ is measurable in $(\Omega', \mathcal F')$, but $X^{-1}(\{1\}) = \{1\}$ is not measurable in $(\Omega, \mathcal F)$.
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space for a fair coin flip $(\{H,T\}, 2^{\Omega}, \mathbb{P}(\omega)=\frac12)$, and let $(\Omega',\mathcal{F}')$ be a (probable?) measurable space for a die roll $(\{1,2,3,4,5,6\}, 2^{\Omega'})$. Let $X(\omega) = 1_{H}(\omega)$, a random variable indicating payoff of flipping heads in the fair coin.T hen $X$ is $\mathcal{F}$-measurable but not $\mathcal{F'}$ measurable.