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''predictable  $  \sigma $-
algebra''

The least  $  \sigma $-
algebra  $  {\mathcal P} = {\mathcal P} ( \mathbf F ) $
of sets in

$$ 
\Omega \times \mathbf R _ {+}  = \ 
\{ {( \omega , t) } : {\omega \in \Omega , t \geq  0 } \}
$$

generated by all mappings  $  ( \omega , t) \rightarrow f ( \omega , t) $
of the set  $  \Omega \times \mathbf R _ {+} $
into  $  \mathbf R $
that are (for each fixed  $  \omega \in \Omega $) 
continuous from the left (in  $  t $) 
and  $  \mathbf F $-
adapted to a non-decreasing family  $  \mathbf F = ( {\mathcal F} _ {t} ) _ {t \geq  0 }  $
of sub- $  \sigma $-
algebras  $  {\mathcal F} _ {t} \subseteq {\mathcal F} $, 
$  t \geq  0 $, 
where  $  ( \Omega , {\mathcal F} ) $
is a measurable space. A predictable  $  \sigma $-
algebra can be generated by any of the following families of sets:

1)  $  A \times \{ 0 \} $, 
where  $  A \in {\mathcal F} _ {0} $
and  $  [[ 0, \tau ]] $, 
where  $  \tau $
is a stopping time (cf. [[Markov moment|Markov moment]]) and  $  [[ 0, \tau ]] $
a [[stochastic interval]];

2)  $  A \times \{ 0 \} $, 
where  $  A \in {\mathcal F} _ {0} $, 
and  $  A \times ( s, t] $, 
where  $  s < t $
and  $  A \in {\mathcal F} _ {s} $.

Between optional  $  \sigma $-
algebras (cf. [[Optional sigma-algebra|Optional sigma-algebra]]) and predictable  $  \sigma $-
algebras there is the relation  $  {\mathcal P} ( \mathbf F ) \subseteq {\mathcal O} ( \mathbf F ) $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Dellacherie,   "Capacités et processus stochastique" , Springer  (1972)</TD></TR></table>

====Comments====
Instead of  "(s-) algebra"  one more often uses ( $  \sigma $-) 
field.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Dellacherie,   P.A. Meyer,   "Probabilities and potential" , '''A-C''' , North-Holland  (1978–1988)  (Translated from French)</TD></TR></table>