Motivation: I am reading this paper where the following space is introduced:
$$\mathcal{D}=\{(y_1,y_2,...):\exists k_0\in\mathbb{N}:\forall k\leq k_0:y_k\in\mathcal{D}([0,T],\mathbb{R}^d)\cup \{\infty\}; \forall k>k_0:y_k=\infty\}$$
Here, $\mathcal{D}([0,T],\mathbb{R}^d)$ is the Skorokhod space of cadlag paths. $\mathcal{D}$ is equipped "with the Skorokhod topology modified in the obvious way to account for the fact that the sample paths have an initial segment $=\infty$."
Now the function
$$F:\mathcal{D}\rightarrow\mathbb{N}\quad (y_k)
\mapsto \sum_{k=1}^\infty 1_{G_k\cap \{y_k(L^2)\in I\}}$$
(where
$$G_k=\{y_k(t)\in[0,4L]^d, 0\leq t\leq L^2\}$$
and
$$I=[L,2L]^d\quad )$$
is claimed to be continuous almost everywhere w.r.t. $\mathbb{P}_Y$ where $Y$ is a Branching Brownian motion. Intuitively, $F(y_1,y_2,...)$ is the number of paths in $(y_1,y_2,...)$ that don't escape $[0,4L]^d$ and end in $I$.
Question 1: What is the "obvious" way to get an appropriate topology on $\mathcal{D}$?
The most "obvious" choice I can see, would be the topology induced by the metric
$$\rho((x_k),(y_k))=\sum_{k=1}^\infty 2^{-k}\frac{\bar{d}(x_k,y_k)}{1+\bar{d}(x_k,y_k)}$$
where $d$ is the Skorokhod metric on $\mathcal{D}([0,T],\mathbb{R}^d)$ and the extended metric on $\mathcal{D}([0,T],\mathbb{R}^d)\cup\{\infty\}$:
$$\bar{d}(x,y)=
\begin{cases}
d(x,y)&x,y\neq \infty\\
0&x=y=\infty\\
\infty&\text{else}
\end{cases}$$
using the convention $\frac{\infty}{1+\infty}=1$.
Question 2: How can one prove the claimed continuity of $F$, i.e. show that for the set $\mathfrak{D}\subset \mathcal{D}$ of discontinuity points of $F$ we have
$$\mathbb{P}(Y\in\mathfrak{D})=0$$
I think that, intuitively, only paths hitting the boundary of $[0,4L]^d$ or ending on the boundary of $[L,2L]^d$ can cause discontinuities, and since these boundaries are sets of measure zero the probability of taking such a path is $0$.
Any help is greatly appreciated!