The hitting number "N""t"("x") for the flow "S""t" on "X" is defined as the integer "N" such that "t" lies in ["r""N"("x"),"r""N"+1("x")). It is an integer-valued Borel function on R × "X" satisfying the cocycle identity  The function "h" = "r"1 is a strictly positive Borel function on Ω so formally the flow can be reconstructed from the transformation "T" using "h" a ceiling function. The missing "T"-invariant measure class on Ω will be recovered using the second cocycle "N""t". Indeed the discrete measure on Z defines a measure class on the product Z × "X" and the flow "S""t" on the second factor extends to a flow on the product given by  Likewise the base transformation "T" induces a transformation "R" on R × Ω defined by  These transformations are related by an invertible Borel isomorphism Φ from R × Ω onto Z × "X" defined by  Its inverse Ψ from Z × "X" onto R × Ω is defined by  Under these maps the flow "R""t" is carried onto translation by "t" on the first factor of R × Ω and, in the other direction, the invertible "R" is carried onto translation by -1 on Z × "X". It suffices to check that the measure class on Z × "X" carries over onto the same measure class as some produce measure "m" × ν on R × Ω, where "m" is Lebesgue measure and ν is a probability measure on Ω with measure class invariant under "T". The measure class on Z × "X" is invariant under R, so defines a measure class on R × Ω, invariant under translation on the first factor. On the other hand the only measure class on R invariant under translation is Lebesgue measure, so the measure class on R × Ω is equivalent to that of "m" × ν for some probability measure on Ω. By construction ν is quasi-invariant under "T". Unravelling this construction, it follows that the original flow is isomorphic to the flow built under the ceiling function "h" for the base transformation "T" on (Ω,ν).