Uniform Mean-Square Continuity on a Compact Interval

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $\mathbb{R}$ be the real numbers, let $a\le b$ be real numbers, and let $(H_t)_{t\in[a,b]}$ be a family of square-integrable random variables that is mean-square continuous on the closed interval $[a,b]$.

Then $(H_t)_{t\in[a,b]}$ is uniformly mean-square continuous: for every real $\varepsilon>0$ there is a real $\delta>0$ such that all $s,t\in[a,b]$ with $|s-t|<\delta$ satisfy $\lVert H_s-H_t\rVert_{2}<\varepsilon$, with $\lVert\cdot\rVert_{2}$ the mean-square norm of Square-Integrable Random Variables and the Mean-Square Inner Product.

Consequently the function $t\mapsto\mathbb{E}[H_t^{2}]$ is bounded on $[a,b]$, and if $a<b$ it is continuous on $[a,b]$, where the domain $[a,b]$ and the codomain $\mathbb{R}$ both carry the metric of the real line.