Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $(E,\|.\|) $ be a Banach space.
$$ L_{E}^{1}(\Omega,\mathcal{F},\mathbb{P})=\{f:\Omega \rightarrow E : \int_{\Omega} \|f\| d\mathbb{P} < \infty\} $$ Let $(X_n)_n \subset L_{E}^{1}( \Omega)$ be a sequence of random variable and $\left(\mathcal{F}_n\right)_{n\ge 1} $ is an increasing sequence of sub- $\sigma$ -algebra of $\mathcal{F}$ , such as : $(X_n,\mathcal{F}_n)_n $ be a martingal. Let $ t\geq 0$ we put: $$ \sigma : \Omega \to \mathbb {N}\cup \{\infty\} $$ Such as : $$ \sigma(\omega)= \begin{cases} n & \text{if } \|X_i(\omega)\|<t ~for ~i=1,...,n-1~and ~\|X_i (\omega)\|\geq t\\ +\infty & \|X_i(\omega)\|<t~\forall i \end{cases} $$ $\sigma$ is a stopping time with respect to $(\mathcal{F}_n)_n$.
Show that $(X_{\sigma\wedge n },\mathcal{F}_n)_n $ is martingal and $(\|X_{\sigma\wedge n }\|,\mathcal{F}_n)_n $ is a sub-martingal.