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Let $X$ be a predictable process defined on some filtered probability space (as good as possible) such that

$$X_t \in \{0,1\},\quad \forall t\ge 0.$$

Does this imply the left continuity of $X$? If so, is there a reference or a proof?

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In the following example, all random variables are constant with respect to $\omega$, so $\Omega$ could be taken to be a point mass.

For each $n$, let $$X_t^n = \begin{cases} 0, & t \le 1-1/n \\ 1, & t > 1-1/n.\end{cases}$$ Then $X_t^n$ is adapted (trivially) and left continuous, so it is predictable. Let $X_t = \lim_{n \to \infty} X_t^n$ which is thus also predictable. But $$X_t = \begin{cases} 0, & t < 1 \\ 1, & t \ge 1\end{cases}$$ which is not left continuous.

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    $\begingroup$ You are absolutely right! Thank you Nate. I love your smart and simple construction $\endgroup$
    – Fawen90
    Commented May 22 at 17:47

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