Stochastic interval

One of the intervals:

$$ \left [ \left [ \sigma , \tau \right ] \right ] = \ \{ {( \omega , t) } : {t \geq 0, \sigma ( \omega ) \leq t \leq \tau ( \omega ) } \} , $$

$$ \left [ \left [ \sigma , \tau \right [ \right [ = \ \{ {( \omega , t) } : {t \geq 0,\ \sigma ( \omega ) \leq t < \tau ( \omega ) } \} , $$

$$ \left ] \left ] \sigma , \tau \right ] \right ] = \ \{ {( \omega , t) } : {t \geq 0,\ \sigma ( \omega ) < t \leq \tau ( \omega ) } \} , $$

$$ \left ] \left ] \sigma , \tau \right [ \right [ = \ \{ {( \omega , t) } : {t \geq 0,\ \sigma ( \omega ) < t < \tau ( \omega ) } \} , $$

where $ \sigma = \sigma ( \omega ) $ and $ \tau = \tau ( \omega ) $ are two stopping times defined on a measurable space $ ( \Omega , {\mathcal F}) $ with an increasing family $ \mathbf F = ( {\mathcal F} _ {t} ) _ {t\geq } 0 $ of sub- $ \sigma $- algebras $ {\mathcal F} _ {t} \subseteq {\mathcal F} $.

References

[1]	C. Dellacherie, "Capacités et processus stochastiques" , Springer (1972)