### 4.6.3. Riez's decomposition

We know that any submartingales can be decomposed into a martingale and a predictable sequence (Doob’s decomposition). Riez’s decomposition allows us to do the similar to uniformly integrable non-negative supermartingales.

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### 4.6.2. Levy's theorem

Consider a sequence of conditional expectations $E(X|\mathcal{F}_n)$ with fixed $X.$ By using the theorem from previous subsection we can determine convergence of this sequence as well.

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### 4.6.1. Uniform integrability and convergence in $L^1$

In section 4.4, we covered the condition where martingales converges in $L^p.$ We only covered the case where $p>1.$ In this section, the notions of uniform integrability is introduced to compensate convergence in $p=1$ case.

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### 4.4. Martingale inequalities and convergence in $L^p$

In this section we look into the condition that makes a martingale converges in $L^p$, $p>1$ in detail. We start by proving *Doob’s inequality*. By using this result we prove *martingale inequalities* which will then be used to prove *Doob’s $L^p$ maximal inequality*. $L^p$ convergence is direct from them. Lastly, as an extension of Doob’s inequality, I will briefly state a version of optional stopping.

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### 4.3. Application of martingales

For applications of martingales, I would like to cover the case of martingales with bounded increments and the branching process.

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