Doob's Maximal Inequality for Square-Integrable Submartingales

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},P)$ be a filtered probability space, let $M=(M_t)_{t\ge0}$ be a square-integrable submartingale with respect to $(\mathcal{F}_t)_{t\ge0}$, let $n$ be zero or a natural number, and let $0\le t_0<t_1<\dots<t_n$ be real numbers. Define the running maximum

the largest of the finitely many values $M_{t_0}(\omega),\dots,M_{t_n}(\omega)$. Write $\mathbf{1}_{A}$ for the function equal to $1$ on $A$ and $0$ off $A$. Then:

1. $M^{*}$ is a square-integrable random variable.

2. For every real $\lambda>0$, the sets $\{M^{*}\ge\lambda\}$ and $\{M^{*}>\lambda\}$ are events, and

the expectations being defined because multiplying the square-integrable (hence integrable) random variable $M_{t_n}$ by an indicator yields an integrable random variable.