Let $(X_i)$ be a simple random walk in $\mathbb{Z}$ starting from the origin. Let $\Sigma$ be the set whose elements are $(Y_i)_{i \in \mathbb{N}}$, which are infinite random walk trajectories. Consider an event $\mathcal{A} \subset \Sigma$. We have that, since we sum over the probability of disjoint events, $$ P(A) = \sum\limits_{(Y_i)_{i \in \mathbb{N} }\in \mathcal{A}} P(X_i = Y_i \, \, \, \forall i \in \mathbb{N} ). $$
Now the sum on the right hand side is uncountable and each probability on the right hand side equals zero. So does the expression make sense? How is it possible to give sense to it? Should I write it as an integral over all possible trajectories?