# Definition of predictable process

I am trying to understand the notion of predictable process. Let $$(Ω,F_t,P)$$ be a filtered measure space, satisfying the usual condition. Things starts with the predictable $$\sigma$$-algebra $${\mathcal P}$$, which is generated by sets of the form $$A\times (a,b]$$ with $$A\in{\mathcal F}_a$$ and $$A\times \{0\}$$ with $$A\in{\mathcal F}_0$$.

My question: is it true that $$S\in {\mathcal P}$$ if and only if $$S$$ is progressive and $$\{\omega|(\omega,t)\in S\}\in{\mathcal F}_{t−}$$ for all $$t$$? In another word, is it true that $$X$$ is predictable if and only if $$X$$ is progressive and $$X$$ is adapted to the filtration $${\mathcal F}_{t−}$$?

The only if part is easy but I am not sure about the if part. I feel that $$X$$ being $${\mathcal F}_{t−}$$-measurable seems to be a more "reasonable" definition of "predictable", but maybe I am wrong.

First of all, recall that any adapted càdlàg process $$(X_t)_{t \geq 0}$$ is progressively measurable. This means that for any such process $$(X_t)_{t \geq 0}$$ your assertion reads
$$(X_t)_{t \geq 0}$$ is predictable $$\iff$$ $$X_{t}$$ is $$\mathcal{F}_{t-}$$-measurable for any $$t \geq 0$$.
Now consider for instance a Poisson process $$(X_t)_{t \geq 0}$$, and let $$(\mathcal{F}_t)_{t \geq 0}$$ be its completed canonical filtration. By the very definition of $$\mathcal{F}_{t-}$$, we know that $$X_{t-} = \lim_{s \uparrow t} X_s$$ is $$\mathcal{F}_{t-}$$-measurable. Since $$X_t = X_{t-}$$ almost surely we find that $$X_t$$ is $$\mathcal{F}_{t-}$$-measurable for any $$t \geq 0$$. However, $$(X_t)_{t \geq 0}$$ is not predictable. Indeed: If $$(X_t)_{t \geq 0}$$ was predictable, then
$$M_t := X_t -t \mathbb{E}(X_1), \qquad t \geq 0,$$
would be a predictable martingale which would imply that $$(M_t)_{t \geq 0}$$ has continuous sample paths (see e.g. here) which is clearly not true; hence $$(X_t)_{t \geq 0}$$ is not predictable.