Left continuity of a process, example

Question

Let X be a bounded $\mathcal F_s-$measurable random variable. Let $s < t$. How can I see that the process $$\big(X\mathbb1_{(s,t]}(u)\big)_{u\ge0}$$ is left-continuous?

Answer 1

A left-continuous process $\ (Y_{u})_{u\ge 0} \ $ is such a process, that almost all its trajectories are left-continuous functions.

More formally:

$$\mathbb{P}\Bigg( \Big\{\omega \in \Omega : \ \ \mathbb{R}_{+}\ni u \ \ \longmapsto \ Y_{u}(\omega) \ \ \ \text{is left-contionous} \ \Big \} \Bigg) =1. \ $$

Fix $\ \omega \in \Omega. \ $In our case, hence $\ X \ $ is bounded, $\ X(\omega)\ $ is equal to some finite value $\ a\in \mathbb{R} \ $ and $$\mathbb{R}_{+} \ni u \longmapsto a \cdot \mathbb{1}_{(s,t]}(u) $$ is obviously left-continuous function.

Lastly note, that $\ X \ $ being $\ \mathcal{F}_{s} \ $ measurable makes the whole process $\Big(X \cdot \mathbb{1}_{(s,t]}(u)\Big)_{u\ge 0}$ adapted to given filtration $\ (\mathcal{F}_{u})_{u\ge 0}$.