examples of random variables that are not measurable

Question

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?

Answer 1

Take $\Omega = \{0,1\}$, $\mathcal{F} = \{\emptyset,\Omega\}$, $\mathbb{P}(\emptyset) = 0$, $\mathbb{P}(\Omega) = 1$, $\Omega'=\Omega$, $\mathcal{F}' = 2^{\Omega}$, and $X(\omega) = \omega$ for every $\omega \in \Omega$. Since $\{1\} \in \mathcal{F}'$ and $X^{-1}(\{1\}) = \{1\} \notin \mathcal{F}$, $X$ is not $\mathcal F/\mathcal F'$-measurable.

For another example, consider $\Omega = [0,1]$, $\mathcal{F} = \{\emptyset, \Omega, [0,\frac{1}{2}],(\frac{1}{2},1]\}$, $\mathbb{P}$ the Lebesgue measure on $[0,1]$, $\Omega'=\Omega$, $\mathcal{F}' = \mathcal{B}(\Omega)$, and $X(\omega) = \omega$ for every $\omega \in \Omega$. Since $[1/4,3/4] \in \mathcal{F}'$ and $X^{-1}([1/4,3/4]) = [1/4,3/4] \notin \mathcal{F}$, $X$ is not $\mathcal F/\mathcal F'$-measurable.

Answer 2

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 $(\Omega, \mathcal F)$.

Answer 3

Let $\Omega=\Omega'$ and let $\mathcal F$ be a proper subcollection of $\mathcal F'$.

Then the identity function $\Omega\to\Omega'$ is not measurable.

Answer 4

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.

Answer 5

Let $(\Omega',\mathcal{F}',\mathbb{P}')$ be a probability space where $\Omega'$ is countable. It can be shown that any $X$ in the probability space must be discrete. Thus, for any non-discrete random variable $Y$ on the probability space $(\Omega,\mathcal{F},\mathbb{P})$, $Y$ is not $\mathcal{F}$-measurable but not $\mathcal{F}'$-measurable.