1) What is the application of Doob's inequality? Can we use Doob's inequality ($L^1$) to prove the convergence (maybe almost surely) of a martingale?
Doob's inequality: Let $X$ be a submartingale taking on non-negative real values, that is, for all times $s$ and $t$ with $s < t$, $E[X_t\mid\mathcal F_s]\geq X_s$. Then, for any constant $C > 0$ and $T>0$ we have
$ \mathbf{P} \left[ \sup_{0 \leq t \leq T} X_{t} \geq C \right] \leq \frac{\mathbf{E} \big[ X_{T} \big]}{C}. $
Doob's inequality ($L^p$): Let $X$ be a martingale,
$S_{t} = \sup_{0 \leq s \leq t} X_{s},\quad\text{for}\ p > 1$
$\| X_{T} \|_{p} \leq \| S_{T} \|_{p} \leq \frac{p}{p-1} \| X_{T} \|_{p}.$
Maybe it is not clear enough. I know that to prove the $L^p$, $p>1$, convergence of a martingale, we can use Doob's inequality in $L^p$ form. However, it seems in the proof of convergence in $L^1$, that Doob's inequality is not used (as far as I understand, it is not enough - we need uniformly integrability).
There are several formly analogous inequalities in probability theory, for example, Komolgorov's inequality, Doob's inequality, and an analogous Doob's inequality in ergodic theory. They all estimate the probability or expectation of a random variable, $|X_n|>M$ where $M$ is a constant. Why are they useful?
2) What is the application for Doob's decomposition theorem? Is it only formly?