21. If is a measurable function of the set to the reals ( including ), then we can write 22. To emphasize this dependency, if f : X \ to Y is a measurable function , we will write 23. The range of a lifting is always a set of measurable functions with the " separation property ". 24. The bounded measurable functions on " X " form a Banach space with respect to the supremum norm. 25. For instance, a "'real-valued measurable function "'is a function for which the preimage of each Borel set is measurable. 26. The Markov transition of the chain is given for any bounded measurable functions " f " by the formula 27. Real numbers in V [ G ] then correspond to Dedekind cuts of such functions, that is, measurable functions . 28. Every injective measurable function from a " standard " probability space to a " standard " measurable space is generating. 29. X \ colon \ Omega \ to E is a measurable function from the set of possible Measure-theoretic definition ). 30. The idea is to first establish the continuous functional calculus then pass to measurable functions via the Riesz-Markov representation theorem.
