I had asked a similar problem before and Didier Piau was very kind to help me with the answer. I have another question on the same problem setting.
Let $T$ be a random exponentially distributed time. $P(T>t)=e^{−t}$. Define $M$ via $M_t=1$ if $t−T∈Q^+$, $M_t=0$ otherwise. Where $Q^+$ being positive rationals. let $F_t$ be a filtration generated by the process $M$.
I can see that the process described above is a martingle but i am trying to prove it is not cadlag. i am trying to apply the definition $\lim_{h\rightarrow 0} M_{t+h} =M_t$ and $\lim_{h\rightarrow 0} M_{t-h} =exists$.
At $M_{t+h}$ i will generate a random variable $T$ and depending on whether $T>t+h$, $M_{t+h}$ will have 0 or 1. In the limit $h\rightarrow 0$ , $M_{t+h}$ does not necessarily go to $M_t$ as at $t+h$ no matter how small $h$ is a new random variable $T$ will decide the value of $M_{t+h}$ which can make $M_{t+h}$ different from $M_t$.
Is my thought process correct to arrive at the conclusion that $M_t$ is not cadlag? It would be very kind of someone to help as I am trying to learn stochastic process on my own reading from a book.