Is a predictable process adapted? Let us consider a measurable space $(\Omega, \mathcal{F})$, with a filtration $(\mathcal{F}_t)_t$ of sub $\sigma$-algebras of $\mathcal{F}$.
The predictable $\sigma$-algebra $\mathcal{P}$ is the $\sigma$-algebra generated by all processes of the form
$$
X : [0,\infty) \times \Omega \rightarrow \mathbb{R}
$$
such that


*

*$X(t,\cdot)$ is $\mathcal{F}_t$-measurable for all $t>0$ (i.e. $X$ is adapted to the filtration $(\mathcal{F}_t)_t$),

*$X(\cdot, \omega)$ is left-continuous for all $\omega \in \Omega$.
My question is:

Is every $\mathcal{P}$-measurable process necessarily adapted? If so, is it necessarily adapted to the coarser filtration $(\mathcal{F}_{t-})_t$?

 A: There is also the concept of an optional process:

Definition 1: The optional $\sigma$-algebra $O$ is generated by all adapted càdlàg processes (continue à droite et limit à gauche/continuous from the right and limit from the left). A stochastic process $X=(X_t)_{t\geq 0}$ is called optional, if $X$ is measurable w.r.t. $O\subseteq \mathcal{P}(\Omega\times \mathbb{R}_{\geq 0})$.

It is possible to show that $P\subseteq O$ ($P$ denotes the predictable $\sigma$-algebra). Thus, every predictable process is optional. For optional process, we have the following result:

Theorem 2: If $X$ is optional, then $X$ is $\mathcal{F}\otimes\mathcal{B}(\mathbb{R}_{\geq 0})-\mathcal{B}(\mathbb{R})$ measurable and the random variable $$X_T 1_{T<\infty}:\Omega\rightarrow \mathbb{R}$$ $$\omega\mapsto X_{T(\omega)}1_{T(\omega)<\infty}$$ is $\mathcal{F}_T-\mathcal{B}(\mathbb{R})$-measurable for any stopping time $T$.

Applying this to the stopping time $T\equiv t$ for $t\geq 0$ gives $X_t=X_T 1_{T<\infty}$ is $\mathcal{F}_T$-measurable. Since $\mathcal{F}_T=\mathcal{F}_t$, we see that a predictable process is adapted to the filtration $(\mathcal{F}_t)_{t\geq 0}$
Regarding your second question, we have the following statement:

Theorem 3: Let $T$ be a stopping time and $X$ a predictable process. Then $X_T 1_{\{T<\infty\}}$ is $\mathcal{F}_{T -}$ measurable.

Applying this to the stopping time $T\equiv t$ yields $X_t=X_t 1_{t<\infty}$ is $\mathcal{F}_{t-}$ measurable for all $t\geq 0$ and a predictable process $X$. 
The proof for theorem 3 can be found in the first edition of "Foundations of modern probability theory" by Olav Kallenberg in lemma 22.3. 
A: I wanted to note another method for proving the first part. I shall only sketch the argument.


*

*Show that the indicator function of every predictable rectangle is progressively measurable. This is very easy to show. Note that the the
predictable rectangles form a $\pi$-system which generate the predictable $\sigma$-algebra.

*Now use the Monotone-Class theorem (see David Williams, Page 37) to show that every bounded predictable process is progressively measurable. (Again very straight-forward)

*Now, if $X$ is predictable $X \wedge n$ is bounded predictable and therefore progressively measurable from the previous step. Now just take
limit $n \to \infty$ to conclude that $X$ is progressively measurable.

*Now use the fact that a progressively measurable process is measurable and adapted. (This too is easy to show).

A: Denote
$ \mathcal{P} = \sigma(\mathcal{I}) = \sigma(\{ (s,t]\times E, 0 \leq s < t, E \in \mathcal{F}_s  \} )$
Consider the map
$ f: (\Omega, \mathcal{F}_{t-}) \to ((0,\infty) \times \Omega , \sigma(\mathcal{I})) $
$ \omega \mapsto (t,\omega) $
This map $f$  is $\mathcal{F}_{t-}-\mathcal{P} $ measurable. For this, notice that $f^{-1}(\mathcal{I}) \subset \mathcal{F}_{t-}$, since for any element in $\mathcal{I}$:
$ f^{-1}( (s,r] \times E  ) = 
    \begin{cases}
        E \quad \text{if } s<t\leq r \\ 
        \emptyset \quad \text{if } t \leq s  \text{ or } t > r  
    \end{cases} \in \mathcal{F}_s \subset \mathcal{F}_{t-} 
    $
Hence
$f^{-1}(\sigma(\mathcal{I})) \subset \mathcal{F}_{t-}$ (see Proving measurability of a function only by checking generating sets).  Finally as by the assumption
$ X : ((0,\infty) \times \Omega, \mathcal{P}) \to (\mathbb{R}, \mathscr{B}(\mathbb{R}))  $
a measurable map, the composition of the two measurable maps
$ X_t = X \circ f : (\Omega, \mathcal{F}_{t-}) \to (\mathbb{R}, \mathscr{B}(\mathbb{R}))$
is $\mathcal{F}_{t-}$ measurable.
