Problem 12: Consider the space R and & = P(R) = the collection of all subsets of R. Define the set function p : E -+ 0, 00] by setting p(A) = the number of integers contained in A for each A E E. Construct from p the outer measure M*(E) = inf EP( E. ) : EC UE, Biec). i=1 i=1 Use the Caratheodory condition to define a measurable set: E C R is measurable, that is, E E M, if for any A C R there holds M* ( A) = M*(AnE) + M*(An Eq). It is known that M is a o-algebra. (a) Determine M. (b) Determine the measure / associated with the outer measure *. That is, find a way to compute M(E) for any E E M