Let $(A^\mathbb{N}, \mathcal{B}(A^\mathbb{N}), \mu)$ be a measure space, where $A^\mathbb{N}$ is a set of one-sided sequences over a finite alphabet $A \subset \mathbb{N}$, $\mathcal{B}(A^\mathbb{N})$ is the Borel sigma-algebra generated by cylinder sets and $\mu$ is a measure with full support (say, Bernoulli or Markov).

Is it possible in this case to construct an analogue of the fat Cantor set, namely a set of positive measure that cannot be represented as a union of an open set and a set of measure zero?

It seems to me that the answer is "no", but I could neither prove it myself nor find any reference.