On the other side, all orbit equivalence maps are additionally required to preserve measures between phases spaces, which significantly restricts the pool of possible orbit equivalences. In this aspect ergodic theory provides finer notions to differentiate flows. To summarize, frameworks of Borel dynamics and ergodic theory are in general positions, and while methods used in these areas are intricately related, there are oftentimes no direct implications between results.

In the measurable case, there is a substantial difference between one dimensional and higher dimensional $ d \ge 2 $ flows. There are continuumly many time change inequivalent $
\mathbb{R} $-flows (see [@MR653094]). In higher dimension the situation is simpler. Two relevant results here are due to D. Rudolph [@MR536948] and J. Feldman [@MR1113569].

\[thm:rudolph-thm\] Any two measure preserving ergodic[^2] $ \mathbb{R}^{d} $-flows, $ d \ge 2 $, are time change equivalent.

\[thm:feldman-thm\] Any two quasi measure preserving ergodic $ \mathbb{R}^{d} $-flows, $ d \ge 2 $, are time change equivalent.

A striking difference of Borel framework was discovered in [@MR2578608] by B. Miller and C. Rosendal, where they proved that all non smooth Borel $ \mathbb{R} $-flows are time change equivalent. In other words, the flexibility of considering orbit equivalences which 