On theories of random variables

We study theories of spaces of random variables: first, we consider random variables with values in the interval [0, 1], then with values in an arbitrary metric structure, generalising Keisler's randomisation of classical structures. We prove preservation and non-preservation results for model theoretic properties under this construction: (i) The randomisation of a stable structure is stable. (ii) The randomisation of a simple unstable structure is not simple. We also prove that in the randomised structure, every type is a Lascar type.