Hey All, In my probability theory class we have just started learning about how a probability space is defined by a sample space (which contains all possible events), events and a measure. We briefly went over the idea of the Power Set, which seems to be the set of all subsets in your sample space. My question is what is the difference between the Power Set and the Sample Space ? aren't they both just a collection of all possible outcomes ? Thanks
The powerset is the sample space of the experiment: 'Of all subsets of X, what is the chance that this subset is one specific subset Y. provided an even distribution' Well, then your chance happens to be 1/|P(X)|.. This is of course nothing special. This is just because the powerset is the defined as the set of all subsets. If you ask, 'what is the chance that you have one specific subset which meets property Q' Then your chance is also simply 1/|{x sub X : Q(x)}| ... The powerset can be defined with the identity property (the property which is always true), which is x = x basically. So we get P(x) = {x sub X : x = x}. Edit: not sure you know 'set builder notation', but if Phi(x) and Theta(x) are formulae in x, such as x > 3, or x ^ 2 = 3 et cetera. See {Phi(x) : Theta(x)} as the set of all x for which both formulae are true.
That's not an example of a sample space and a powerset, that's A: a set, and B: the power set of that set. If A is a sample space, then B is the powerset of a sample space. A sample space is with respect to some experiment.
When I learned probability theory, it was taught on an abstract basis. A sample space was defined on the basis of Kolmogoroff axioms and did not require a particular experiment. In essence a sample space was a measure space with total measure = 1.
Yes, all and well, the first set was a possible sample space. But so was the second. You just gave a sample space, and the powerset of that space as an illustration that powerspaces and sample sets are some-how different concepts, which they are, but not via this logic, because a powerset is of course always different to its set.