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Let $(\Omega,\mathcal{F})$ be a measurable space, carrying a stochastic process $X=X_{t≥0}$ with state space $(\mathbb{R},\mathcal{B}(\mathbb{R}))$. Let $\mathcal{F}_t = \sigma(X_s:s≤t)$. Assume that trajectories $t\mapsto X_t(w)$ are continuous for all $w\in\Omega$. Prove or disprove : $(\mathcal{F}_t)$ is right continuous.

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