Let $P = (\Omega, \mathcal{F}_{\Omega}, \mathbb{P})$ be a D1159: Probability space such that
(i) | $M = (\Xi, \mathcal{F}_{\Xi})$ is a D1108: Measurable space |
(ii) | $X : \Omega \to \Xi$ is a D202: Random variable from $P$ to $M$ |
(iii) | \begin{equation} \exists \, \xi \in \Xi : \forall \, \omega \in \Omega : X(\omega) = \xi \end{equation} |
(iv) | $\sigma_{\text{pullback}} \langle X \rangle$ is a D1730: Pullback sigma-algebra for $X$ on $P$ |
Then
\begin{equation}
\sigma_{\text{pullback}} \langle X \rangle
= \{ \emptyset, \Omega \}
\end{equation}