I'm reading a proof of $L^p$ maximal inequality from these notes. In the proof, we have $\left\|X_n^*\right\|_p^p \le C_p^p\left\|X_n\right\|_p \left\|X_n^*\right\|_p^{p / q}$ after Hölder's inequality. The author then makes a simplification to get $\left\|X_n^*\right\|_p^{p-p / q} \leq C_p\left\|X_n\right\|_p$. However, this step would only be possible if $\left\|X_n^*\right\|_p < +\infty$, but it's not obvious to me.
Could you explain how such simplification is valid?
Everything that follows takes place on a probability space $(\Omega, \mathcal{F}, P)$ equipped with a filtration $\left\{\mathcal{F}_n: n=0,1,2, \ldots\right\}$, with $\mathcal{F}_n \subset \mathcal{F}$ for all $n$.
1. Submartingale maximal inequality. Let $\left\{X_n\right\}$ be a non-negative submartingale (for example, $X_n=\left|M_n\right|$ if $\left\{M_n\right\}$ is a martingale, or $X_n=S_n^{+}$if $\left\{S_n\right\}$ is a submartingale), and define $X_n^*:=\max _{0 \leq k \leq n} X_k$. Then $$ P\left[X_n^* \geq t\right] \leq t^{-1} E\left[X_n ; X_n^* \geq b\right] \leq t^{-1} E\left[X_n\right], \quad \forall t>0 . $$
4. Lemma. Let $W$ and $Z$ be non-negative random variables. Then for any $r>0$, $$ E\left[W \cdot Z^r\right]=r \int_0^{\infty} t^{r-1} E[W ; Z>t] d t. $$
5. $L^p$ Maximal Inequality. If $\left\{X_n\right\}$ is a positive submartingale and $1<p<\infty$, then for $n=0,1,2, \ldots$ $$ \left\|X_n^*\right\|_p \leq C_p\left\|X_n\right\|_p, $$ where $X_n^*:=\max _{0 \leq k \leq n} X_k$ and $C_p:=p /(p-1)$.
Proof. Fix $n$. By Lemma 4 (twice) and the maximal inequality 1, $$ \begin{aligned} E\left[\left(X_n^*\right)^p\right] & =p \int_0^{\infty} t^{p-1} P\left[X_n^*>t\right] d t \\ & \leq p \int_0^{\infty} t^{p-2} P\left[X_n ; X_n^*>t\right] d t \\ & =\frac{p}{p-1} E\left[X_n\left(X_n^*\right)^{p-1}\right]. \end{aligned} $$
Thus, by Hölder's inequality, $$ (1) \quad \left\|X_n^*\right\|_p^p = E\left[\left(X_n^*\right)^p\right] \leq \frac{p}{p-1} E\left[X_n\left(X_n^*\right)^{p-1}\right] \leq C_p^p\left\|X_n\right\|_p \cdot\left\|\left(X_n^*\right)^{p-1}\right\|_q . $$ Here $q=p /(p-1)$ is the conjugate exponent of $p$. In particular, $(p-1) q=p$, so $\left\|\left(X_n^*\right)^{p-1}\right\|_q=\left\|X_n^*\right\|_p^{p / q}$. Therefore, (1) implies $$ \left\|X_n^*\right\|_p^{p-p / q} \leq C_p\left\|X_n\right\|_p, $$ which is the stated inequality because $p-p / q=1$.
You are correct that there is a gap in the proof. However, we can apply the same proof with $X_n^*$ replaced by $X_n^* \wedge k$ to get $\|(X_n^* \wedge k)\|_p \le C_p \|X_n\|_p$ for all $k$. Sending $k$ to $\infty$ and applying the monotone convergence theorem then gives $\|X_n^*\|_p \le C_p \|X_n\|_p$.