I know the supremum is measurable if \Theta is countable. So maybe this is what we should start with ? Any comments and answers are appreciated
Thank your very much Joris ! Indeed, \forall \theta f is at least a piecewise continuous function in x and therefore Borel measurable; and for almost all x f is continuous in \theta. Also \Theta is an abstract space which is absolutely non-euclidiean. So I have tried to find an isomorphic mapping from \Theta to a subset of a euclidean space, but this does not seem trivival (\Theta is equipped with a norm). So I find your first suggestion very interesting. But do you mean that we can always use this reasoning given continuity of f in x, why did you write "may be". Which role does "pi-system" play here? I am so sorry but I am not so farmliar with "pi-system".
Greatly thank again Joris ! I think I have just proved that \Theta is a subset of a separable Banach space. And we can choose \Theta as an open subset ! So following your suggestion I think I can show the measurability. Nevertheless, I think the compactness is still needed as I need not only the sup but actually the arg max, the value \theta_0 such that f(x,\theta_0) = sup ....., to be measurable . But this is another story. Again I really appreciate your help. I hope we can still keep in touch in the future !
If your function is continuous on theta for *all* x (as well as measurable on x for all \theta) then it is called a Carathéodory function. Since you say it works well for all \theta, maybe you can adapt results for Carathéodory functions by dropping the null set of bad-behaved x and taking some care with the completeness of the sigma-algebras (to avoid introducing non-measurable sets).
For a Carathéodory function, you have all you need: both the supremum and the argmax are measurable. See, for instance, the "measurable maximum theorem" in the "Infinite Dimensional Analysis: A Hitchhiker's Guide", which you can consult at Google Books. It is written in a more general language for multifunctions, but the assumption is that the values of the multifunction are compact; that is trivial in your case since each value of your function is a single point.
If you cannot make it with the Carathéodory function approach, there are also results which do not require continuity. But joint measurability in both arguments of your function is needed, as far as the results I am aware of go (a Carathéodory function is always jointly measurable). In any case you may be able to prove joint measurability from the fact that your function is "almost" a Carathéodory function.
For instance, see Proposition 4.2 in the paper attached below.
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On the other hand, when dealing with the argmax, do you have a unique argmax?
http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.aos/1032526970
Thank you so much Pedro ! Never thought that my question'sgot so much interest ! I will give it a try with your suggestion.
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