Let $A$ be a set that is possibly uncountable. Suppose that for each $\alpha \in A$, we associate to it a random variable $X_{\alpha}: \Omega \to \mathbb{R}$. Suppose there exists $x, \eta \in \mathbb{R}$ such that that for all $\alpha \in A$:
$\Pr(X_{\alpha} > x) < \eta$.
Let's define the random variable $\underset{\alpha \in A}{\sup} X_{\alpha}: \Omega \to \mathbb{R}$ by the rule $\left(\underset{\alpha \in A}{\sup} X_{\alpha}\right)(\omega) := \underset{\alpha \in A}{\sup} (X_{\alpha}(\omega))$ for all $\omega \in \Omega$. Does it follow that
$\Pr\left(\underset{\alpha \in A}{\sup} X_{\alpha} > x\right) < \eta$?
For ensuring that $\underset{\alpha \in A}{\sup} X_{\alpha}$ is well-defined, let's suppose that $X_{\alpha}$ are all bounded random variables.