Doob's L2 Maximal Inequality for Bounded Right-Continuous Martingales on a Compact Time Interval

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $T>0$ and $K\ge0$ be real numbers, let $(\mathcal{F}_t)_{t\in[0,T]}$ be a filtration with time index restricted to $[0,T]$, and let $(M_t)_{t\in[0,T]}$ be a square-integrable martingale with respect to $(\mathcal{F}_t)_{t\in[0,T]}$, with time index restricted to $[0,T]$.

Assume there is an event $\Omega_0\in\mathcal{F}$ with $P(\Omega_0)=1$ such that for every $\omega\in\Omega_0$ the path $t\mapsto M_t(\omega)$ satisfies $|M_t(\omega)|\le K$ for every $t\in[0,T]$ and is right-continuous at every $t\in[0,T)$, in the sense of the supremum lemma for bounded right-continuous processes, and let $D$ and $\overline{M}=\sup_{t\in D}\big(|M_t|\,\mathbf{1}_{\Omega_0}\big)$ be as in that lemma, where $\mathbf{1}_{\Omega_0}$ is the function equal to $1$ on $\Omega_0$ and $0$ elsewhere.

Then

$$\mathbb{E}\big[\overline{M}^{\,2}\big]\le4\,\mathbb{E}\big[M_T^2\big],$$

and consequently $\mathbb{E}\big[\sup_{t\in[0,T]}|M_t|^2\big]\le4\,\mathbb{E}[M_T^2]$, the supremum being taken pathwise on $\Omega_0$.