Progressive measurability implies adaptedness I've read that every progressively measurable process is also adapted, but I can't prove it using the definition of measurability.
Can anyone give me a proof of this result ?
 A: It seems to me that this fact is a great example of something that is evident to someone experienced, but in fact confuses a great many students by being labeled as such with no explanation, myself included.
First, let's lay down some definitions. Let $(\Omega,\mathcal F, P)$ be a probability space and $\mathcal F_t$ a filtration on it.

The process $(X_t)_{t\geq 0}$ is

*

*adapted if $X_t:\Omega \to \mathbb R$ is $\mathcal F_t$ measurable for each $t\geq 0$

*progressively measurable if $X:[0,t]\times \Omega \to \mathbb R$ is $\mathcal B([0,t])\otimes \mathcal F_t$ measurable for each $t>0$

In terms of sets, this means

*

*Adapted: $\{\omega: X_t(\omega) \in B\}\in \mathcal F_t$ for all, $t\geq 0 , B\in \mathcal B(\mathbb R)$:

*Progressively measurable: $\{ (s,\omega): X_s(\omega) \in B \} \in \mathcal B([0,t])\otimes \mathcal F_t$ for all $t>0, B \in \mathcal B(\mathbb R)$
The key realization is that the set $\{\omega: X_t(\omega) \in B\}$ is, in fact, a cross-section (or slice) of the set $\{ (s,\omega): X_s(\omega)\in B\}$ at the point $t$, which yields the implication progressively measurable $\Rightarrow$ adapted.
The measurability of cross-sections is typically introduced in measure theory courses/books when introducing product measures and can be shown by a standard measure theoretic argument.
