I'm reading some lecture notes about time series where the author mentions measurable functions from $\mathbb R^\infty$ to $\mathbb R$. I was never introduced to $\mathbb R^\infty$, so I have a few questions:
What's the standard meaning of $\mathbb R^\infty$ ? Is it the space of sequences of real numbers ($\mathbb R^{\mathbb N})$?
What's the usual sigma-algebra on $\mathbb R^\infty$ ? I guess it is the Borel sigma-algebra, but one needs to specify a metric or a topology in the first place. What's the most natural/common metric/topology used ?
Here's the context in which $\mathbb R^\infty$ is used: Let $(Z_t)_{t\in \mathbb Z}$ be a strictly stationary, ergodic process and $f:\mathbb R^\infty \to \mathbb R^d$ be measurable. Let $Y_t=f(Z_t, Z_{t-1},\ldots)$. Then $(Y_t)_{t\in \mathbb Z}$ is strictly stationary and ergodic.
In this context, $\mathbb R^{\infty}$ is being used as the space of all maps $\mathbb N\rightarrow\mathbb R$. In categorical terms, the author here wants to use the direct product (in the category of measurable spaces) of one copy of $\mathbb R$ for each natural number - since this is the right way to make the expression $f(Z_t,Z_{t-1},\ldots)$ make sense if $f$ is a measurable map $\mathbb R^{\infty}\rightarrow\mathbb R^d$.
Explicitly, the $\sigma$-algebra on $\mathbb R^{\infty}$ is the one generated by sets of the form $\{x\in \mathbb R^{\infty}:x_i\in U\}$ where $i\in \mathbb N$ and $U\subseteq \mathbb R$ is open.