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A complete probability space  $  ( \Omega , {\mathcal F} , {\mathsf P}) $
with an increasing family  $  \mathbf F = ( {\mathcal F} _ {t} ) _ {t \geq  0 }  $
of sub- $  \sigma $-
algebras  $  {\mathcal F} _ {t} \subseteq {\mathcal F} $, 
which satisfies the (so-called usual) conditions:

1) it must be continuous from the right,  $  {\mathcal F} _ {t} = {\mathcal F} _ {t  ^ {+}  } $(
$  = \cap _ {s>} t {\mathcal F} _ {s} $), 
$  t \geq  0 $;

2) it must be complete, i.e.  $  {\mathcal F} _ {t} $
contains all subsets from  $  {\mathcal F} $
of  $  {\mathsf P} $-
measure zero.

For stochastic bases, the notations  $  ( \Omega , {\mathcal F}, \mathbf F , {\mathsf P}) $
or  $  ( \Omega , {\mathcal F} , ( {\mathcal F} _ {t} ) _ {t \geq  0 }  , {\mathsf P}) $
are also used.

====Comments====
An increasing family of ( $  \sigma $-) 
algebras is usually called a filtration.