Let $X$ be a separable metric space, $\phi\in C(X,X)$ be a topologically transitive dynamical system, and $V$ be a non-empty open subset of $X$, and $\nu$ be a locally-positive and atomless Borel probability measure on $X$.

Then, for every $\delta \in (0,1)$, does there exist:

$\{V_i\}_{i \in \mathbb{N}}$ are open subsets of $V$ satisfying $\nu\left( V-\cup_{n \in \mathbb{N}} V_n \right)=0$

A sequence $\{N_i\}_{i \in \mathbb{N}}$,

*such that the following holds:*$$ \nu\left(X - \cup_{i \in \mathbb{N}} \phi^{-N_i}[V_i] \right)<\delta . $$

*In intuitive words: there exists an open cover of a non-empty open set be reversed into an almost-everywhere cover of the entire space by appropriately reversing the dynamical sytem?*

**Auxiliary Definitions:**

A Borel measure $\nu$ on $X$ is said to be

*locally-positive*iff for every non-empty open subset $U\subseteq X$, $\nu(U)>0$.*For example, if $X$ has more than two points then the Dirac is not such a measure.*$\phi$ is said to be

*topologically transitive*iff for every two non-empty open subsets $U,V\subseteq X$ there exists some $N\in \mathbb{N}$ such that $$ \phi^N(U)\cap V \neq \emptyset. $$