Maximum of submartingales

Question

Let $\left\{ X_t\right\}_{t\in\mathbb{T}}$, $\left\{ Y_t\right\}_{t\in\mathbb{T}}$ be submartingales with the respect to the filtration $\left\{\mathcal{F}_t\right\}_{t\in\mathbb{T}}$. Show, that $\left\{\max\left\{X_t,Y_t\right\}\right\}_{t\in\mathbb{T}}$ is also the submartingale with the respect to $\left\{\mathcal{F}_t\right\}_{t\in\mathbb{T}}$.

Directly from the definition we have that $\left\{\max\left\{X_t,Y_t\right\}\right\}=X_t\chi(X_t\ge Y_t)+Y_t\chi(X_t\le Y_t)$ and now if I show that $\chi(X_t\ge Y_t)$ and $\chi(X_t\le Y_t)$ are measurable (with the respect to $\left\{\mathcal{F}_s\right\}$), I'll got the third condition from definition of submartingales. My problem is showing above measurability. Any hint?

Answer 1

Note $X_t\ge Y_t \Leftrightarrow X_t-Y_t\ge0$ . Since $X_t,Y_t\in\mathcal{F}_t$ for all $t$, $X_t-Y_t\in\mathcal{F}_t$. Since $[0,\infty)$ is a Borel set, $\{(X_t-Y_t)\in [0,\infty)\}\in\mathcal{F}_t$. Hence the indicator function of the set is $\mathcal{F}_t$-measurable.