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Random variable X (as a measureable function from some probability space to the reals) has support S and image T. True or False:

i.) S is always a subset of T.

ii.) (S,E,dF) is the induced probability space with: sample space S, events E the intersections of S with the Borel sets, and dF the probability measure derived from F -the distribution of X.

(As S is the smallest closed subset of reals R with probability 1, it is a subset of the closure of T. The two issues come from having read somewhere that the support is, "loosely" speaking, the outcomes X can take on mapping the space to R. Started with case T finite.)

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    What do you mean by *induced probability space* in ii)?2012-09-14
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    I asked this question as "ace091312". Thanks Rabee and Michael for your answers settling the issue. They actually led to another related question (key words: support, image and image closure) which I posted posted under "ace" yesterday.2012-09-17

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