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Oct 19, 2022 at 21:16 comment added Michael Greinecker Atomless probability spaces where this is possible a very special. The main examples are uncountable products of the uniform measure and hyperfinite Loeb spaces. The book "Model Theory of Stochastic Processes" by Fajardo and Keisler discusses the property in detail.
Oct 19, 2022 at 20:38 answer added Martin Hairer timeline score: 4
Oct 19, 2022 at 19:53 comment added Christian Remling Of course, this cannot work pointwise, so you probably wanted the equality to hold a.e. on $[0,1]$ ?
Oct 19, 2022 at 8:20 history asked daon CC BY-SA 4.0