Let $X$ be a compact metric space and let $T: X \rightarrow X$ be continuous. Let $\mu$ be a $T$-invariant Borel probability measure (which we can always find by the Krylov-Bogoliubov theorem).
Let $(U_j)_{j \in [0, 1]}$ be a collection of distinct Borel sets with $0 < \mu(U_j)< \infty$ for each $j \in [0, 1]$. Suppose that for any non-empty $I \subseteq [0, 1]$ we have:
$$\ \ \mu\left(\bigcap_{i \in I} U_i \cap \bigcap_{k \in [0, 1]\backslash I} U^\mathsf{c}_k \right) = 0. \tag{1}$$
And suppose:
$$ U_j \subseteq \bigcup_{I \in \mathcal{P}([0, 1])\backslash \emptyset} \left(\bigcap_{i \in I} U_i \cap \bigcap_{k \in [0, 1]\backslash I} U^\mathsf{c}_k \right), \mbox{ for any } j \in [0, 1]. \tag{2}$$
Questions: is it always true that some such collection $(U_j)_{j \in [0, 1]}$ exists? If so, can we say anything general about its properties?
Thoughts: this is easy for specific choice of $X$ and $T$. Consider e.g. $X = [0, 1]$ with the Lebesgue measure and take $T$ to be the identity map. Then $U_x = \{y: x - 1/2 \leq y \leq x + 1/2\} \cap [0, 1]$, $x \in [0, 1]$ is one such collection. Is the condition (2) trivially satisfied if (1) holds?
