Questions on Doob's Optional Stopping Theorem (a) and (b)

Question

From Williams' Probability w/ Martingales:

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1. What is $X_T$ in red box above? I am fairly certain this was not defined previously in the book. There was this though:

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I have a feeling $X_T = \lim_{n \to \infty} X_{T \wedge n}$ based on what follows (red boxes below):

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2. In proof (a), how do we know $X_{T \wedge n}$ is integrable? I guess $E[X_{T \wedge n}] (\le E[X_0]) < \infty$ so it has finite expectation, but how do we know that $E[|X_{T \wedge n}|] < \infty$?

3. For (a), how do we know $X_T$ is integrable? Assuming my conjecture in (1) is right, I guess:

$$X_T = \lim_{n \to \infty} X_{T \wedge n}$$

$$\to |X_T| = |\lim_{n \to \infty} X_{T \wedge n}|$$

$$\to |X_T| = \lim_{n \to \infty} |X_{T \wedge n}|$$

$$\to E[|X_T|] = E[\lim_{n \to \infty} |X_{T \wedge n}|]$$

$$\to E[|X_T|] ?=? \lim_{n \to \infty} E[|X_{T \wedge n}|] $$ for some reason. Dominated convergence theorem using $X_0$ ?

If so $E[|X_{T \wedge n}|]$ is a number and hence $E[|X_T|] < \infty$

4. How to prove (b)?

Do I simply apply (a) to say that

$E[X_T] \le E[X_0]$ and $E[X_T] \ge E[X_0] \to E[X_T] = E[X_0]$?

Answer 1

1. $X_T$ is the random variable with value $X_{T(\omega)}(\omega)$ at each point $\omega\in\Omega$ where $T(\omega)<\infty$. For $\omega\in\{T=\infty\}$ one can take $X_T(\omega)$ to equal $\lim\limits_{n\to\infty}X_n(\omega)$ for those $\omega$ for which the limit exists.

2. $|X_{T\wedge n}|=\sum\limits_{k=0}^{n-1}1_{\{T=k\}}|X_k|+1_{T\ge n}|X_n|\le \sum\limits_{k=0}^n|X_n|$, and this is a finite sum of integrable random variables.

3. The proof of integrability of $X_T$ depends on which of the three cases (i),(ii),(iii) is under consideration.

4. Yes. For (b), apply (a) to $X$ and to $-X$.