Let T be any set and (Ω, A, P) a probability space. Recall that a real-valued stochastic process indexed by T is a function (t, ω) ↦ Xt(ω) from T × Ω into ℝ such that for each t ∈ T, Xt (·) is measurable from Ω into ℝ. A modification of the process is another stochastic process Yt defined for the same T and Ω such that for each t, we have P(Xt = Yt) = 1. A version of the process Xt, is a process Zt, t ∈ T, for the same T but defined on a possibly different probability space (Ω1, B, Q) such that Xt and Zt, have the same laws, that is, for each finite subset F of Clearly, any modification of a process is also a version of the process, but a version, even if on the same probability space, may not be a modification. For example, for an isonormal process L on a Hilbert space H, the process M(x) ≔ L(−x) is a version, but not a modification, of L.

One may take a version or modification of a process in order to get better properties such as continuity. It turns out that for the isonormal process on subsets of Hilbert space, what can be done with a version can also be done by a modification, as follows.

TheoremLet L be an isonormal process restricted to a subset C of Hilbert space. For each of the following two properties, if there exists a version M of L with the property, there also is a modification N with the property.