When does an uncountable collection of random variables define a stochastic process?

Question

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $( \mathcal{X}, \mathcal{B})$ be a measurable space. Let $\{X_t\}_{t\in [0,1]}$ be an uncountable collection of random variables such that $$X_t:(\Omega, \mathcal{F}, \mathbb{P}) \to ( \mathcal{X}, \mathcal{B})$$

When does $\{X_t\}$ define a stochastic process? For instance, if $X_t$ are iid standard normal random variables, then the corresponding collection doesn't define a stochastic process, (though I am not sure why).

---

This is an exercise from Lectures on Gaussian processes by Lifshits

Ex 1.3: Let $\mathcal{X}$ be an infinite dimensional separable Hilbert space and $E:\mathcal{X} \to \mathcal{X}$ be the identity operator. Prove that the distribution $\mathcal{N}(0, E)$ does not exist. Hint: If $X$ is a random vector with distribution $\mathcal{N}(0, E)$, it would satisfy an absurd identity $\mathbb{P}(||X||^2 = \infty) = 1$.

Here $\mathcal{N}(0, E)$ refers to the Gaussian process with identity covariance kernel and $0$ mean.

Answer 1

Stochastic process is a collection of random variables — nothing more and nothing less. That means your collection of iid standard normal random variables is an example of stochastic process that even has a special name — it's called a white noise.

Answer 2

The only requirement is that the random variables $X_t$ are defined on the same probability space.