Random variables partitioning a space into chunks over which they are constant $\iff$ Capturing all information provided by the random variable. Cf: [[Statistical Inference#Minimality and Sufficiency]] [[Conditional Expectations]] Consequently, the $\sigma$-algebra created by such partition represents the **information** carried by the RV. - In particular, the trivial $\sigma$-algebra $\{ \emptyset, \Omega \}$ provides no information: for any RV $X$ over $(\Omega, \mathcal{F}, \mathbb{P})$, $\mathbb{E}[X \,|\, \{ \emptyset, \Omega \}] = \mathbb{E}[X].$