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A filtration $\{\mathcal{F}_n\}_{n \geq 0}$ is an increasing sequence of $\sigma$-algebras, meaning that $\mathcal{F}_n \subseteq \mathcal{F}_{n+1}$ for all $n$.

A stopping time with respect to a filtration $\{\mathcal{F}_n\}$ is a random variable $\tau$ such that for each $n$, the event $\{\tau = n\}$ is $\mathcal{F}_n$-measurable.

A sequence $\{X_n\}$ is a supermartingale if $E[|X_n|] < \infty$, $X_n$ is $\mathcal{F}_n$-measurable, and $E[X_{n+1} | \mathcal{F}_n] \leq X_n$ a.s.

A family of random variables $\{X_n\}$ is uniformly integrable if for every $\epsilon > 0$ there exists a $K > 0$ such that $E[|X_n| \mathbf{1}_{\{|X_n| > K\}}] < \epsilon$ for all $n$.

Under certain conditions, if $\{X_n\}$ is a martingale and $\tau$ is a bounded stopping time, then $E[X_{\tau}] = E[X_0]$.

Given a martingale $\{X_n\}$ and a predictable sequence $\{H_n\}$, the martingale transform $\{Y_n\}$ is defined by $Y_n = \sum_{i=0}^{n} H_i \cdot (X_{i+1} - X_i)$, where $Y_n$ is also a martingale under certain conditions.

A martingale $\{X_n\}$ converges in $L^p$ (for $1 < p < \infty$) to a random variable $X$ if $||X_n - X||_p \to 0$ as $n \to \infty$, often requiring $\sup_n E[|X_n|^p] < \infty$.

A sequence $\{X_n\}$ is a submartingale if $E[|X_n|] < \infty$, $X_n$ is $\mathcal{F}_n$-measurable, and $E[X_{n+1} | \mathcal{F}_n] \geq X_n$ a.s.

If $\{X_n\}$ is a martingale (or a sub/supermartingale), then $X_n$ converges almost surely to some random variable $X$ as $n \to \infty$, provided that the martingale is uniformly integrable.