Let $(X_n)_{n \geq 0}$ be an i.i.d. sequence of $\{0,1\}$-valued random variables $X_n \sim \mathrm{Bernoulli}(\frac{1}{2})$, i.e. a sequence of independent tosses of a fair coin.
Does there exist a (non-random) Borel-measurable function $h \colon \{0,1\}^{\mathbb{Z}_{\geq 0}} \to \mathbb{Z}_{\geq 0}$ such that $$ \mathbb{P}(\exists\,\text{infinitely many } k \geq 0 \text{ s.t. }h((X_{k+n})_{n \geq 0})=k ) = 1 \, \text{?} $$
Such a function $h$ does not exist.
Indeed, given a Borel-measurable function $h \colon \{0,1\}^{\mathbb{Z}_{\geq 0}} \to \mathbb{Z}_{\geq 0},$ define for each $k \ge 0$ the events $$A_k=\Bigl\{h \Bigl((X_{n})_{n \geq 0}\Bigr)=k \Bigr\}$$ and $$B_k=\Bigl\{h \Bigl((X_{k+n})_{n \geq 0}\Bigr)=k \Bigr\} \,.$$ Clearly, $$\mathbb{P}(A_k)=\mathbb{P}(B_k) \;\; \mbox{for every} \; k \ge 0. $$ The $A_k$ are disjoint events that partition the probabiility space, so $$ 1=\sum_{k \ge 0} \mathbb{P}(A_k)=\sum_{k \ge 0}\mathbb{P}(B_k) \,. $$ Therefore, by the (first) Borel-Cantelli lemma, $$ \mathbb{P}(\exists\,\text{infinitely many } k \geq 0 \text{ s.t. }h((X_{k+n})_{n \geq 0})=k ) = 0 \,. $$
