Which or these statements is true and what is the rule for continuous space:
$(\mathbb{R},\mathcal{B}(\mathbb{R}),\mathbb{P})$, where $Z = \{ Z_t: 0\leq t\leq 10 \}$:
1) $Z_t(\omega) := t*\omega$
- Z has continuous sample path
2) $Z_t(\omega) := I_{[t,\infty]}(\omega)$
Z has continuous sample path
Z has right continuous sample path
3) $Z_t(\omega) := I_{[-t,t]}(\omega)$
Z has continuous sample path
all the sample paths of Z are continuous (does this paraphrased sentence makes difference)
4) $Z_t(\omega) := t*I_{[-1,1]}(\omega)$
Z has continuous sample path
all the sample paths of Z are continuous (does this paraphrased sentence makes difference)