The Square of a Square-Integrable Martingale with Finite Fourth Moments is a Nonnegative Submartingale

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $T>0$ be a real number, let $(\mathcal{F}_t)_{t\in[0,T]}$ be a filtration on $(\Omega,\mathcal{F},P)$ with time index restricted to $[0,T]$, and let $M=(M_t)_{t\in[0,T]}$ be a square-integrable martingale with respect to $(\mathcal{F}_t)_{t\in[0,T]}$, with time index restricted to $[0,T]$, such that for every $t\in[0,T]$ the random variable $M_t^2$ is itself square-integrable, that is, $\mathbb{E}[M_t^4]<\infty$, where $\mathbb{E}$ is the expectation. Write $\mathbf{1}_{A}$ for the function equal to $1$ on $A$ and $0$ off $A$.

Then the family $M^2=(M_t^2)_{t\in[0,T]}$ is a square-integrable submartingale with respect to $(\mathcal{F}_t)_{t\in[0,T]}$, with time index restricted to $[0,T]$, with $M_t^2(\omega)\ge0$ for every $t$ and $\omega$, and more precisely, for all $0\le s\le t\le T$ and every $A\in\mathcal{F}_s$,

$$\mathbb{E}\big[M_t^2\mathbf{1}_{A}\big]-\mathbb{E}\big[M_s^2\mathbf{1}_{A}\big]=\mathbb{E}\big[(M_t-M_s)^2\mathbf{1}_{A}\big]\ \ge\ 0 .$$