A Generalised Jensen Inequality

In Section 4, we require a version of Jensen's inequality generalised to (possibly) infinite-dimensional vector spaces, because our random variable takes values in H R. Note that this square norm function is indeed convex, since, for any t [0, 1] and any pair f, g H Suppose T is a real Hausdorff locally convex (possibly infinite-dimensional) linear topological space, and let C be a closed convex subset of T. Suppose (Ω, F, P) is a probability space, and V: Ω T a Pettis-integrable random variable such that V (Ω) C. Let f: C [,) be a convex, lower semi-continuous extended-real-valued function such that E We will actually apply generalised Jensen's inequality with conditional expectations, so we need the following theorem. Suppose T is a real Hausdorff locally convex (possibly infinite-dimensional) linear topological space, and let C be a closed convex subset of T. Suppose (Ω, F, P) is a probability space, and V: Ω T a Pettis-integrable random variable such that V (Ω) C. Let f: C [,) be a convex, lower semi-continuous extended-realvalued function such that E Here, (*) and (**) use the properties of conditional expectation of vector-valued random variables given in [12, pp.45-46, Properties 43 and 40 respectively]. The right-hand side is clearly E-measurable, since we have a linear operator on an E-measurable random variable. Now take the supremum of the right-hand side over Q. Then (5) tells us that E [ f(V) | E ] ( f E [ V | E ]), as required.