Let $(X_n)_n$ be a martingale for a filtration $(\mathcal{F}_n)_n$ and let $(R_n)_n$ be a non decreasing sequence of stopping times for $(\mathcal{F}_n)_n,$ a.s finite. Suppose that $\forall n \in \mathbb{N},E[|X_{R_n}|]<+\infty,\liminf_k\int_{R_n \geq k}|X_k|dP=0.$
Prove that $(X_{R_n})_n$ is a martingale for $(\mathcal{F}_{R_n})_n.$ Is there an extension for this result, in the continuous case?
To prove it, it is easy to see that $E[X_{R_n}]<+\infty,$ and that $X_{R_n}$ is $\mathcal{F}_{R_n}$-measurable. It remains to prove that $\forall n,\forall K \in \mathcal{F}_{R_{n}},\int_{K}X_{R_{n+1}}=\int_{K}X_{R_n}dP.$
So how can we prove it? Also, the result remains true for continuous martingale?