Mean-Square Continuous Family of Random Variables

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $I$ be a nonempty set of real numbers, and let $(H_t)_{t\in I}$ be a family of square-integrable random variables on $(\Omega,\mathcal{F},P)$, with $\lVert\cdot\rVert_{2}$ the mean-square norm of that definition.

The family $(H_t)_{t\in I}$ is mean-square continuous at a point $u\in I$ if for every real $\varepsilon>0$ there is a real $\delta>0$ such that every $v\in I$ with $|v-u|<\delta$ satisfies

The family $(H_t)_{t\in I}$ is mean-square continuous on $I$ if it is mean-square continuous at every point of $I$.

When $I$ consists of nonnegative real numbers and $(H_t)_{t\in I}$ is (the restriction to $I$ of) a stochastic process, a mean-square continuous family is also called a mean-square continuous process.