Suppose (En) is a disjoint collection of measurable sets. Use induction on n to establish the following result. for every ΜΕΝ, (A) (An E) + A* E.) + X* (A (n)) for every ACR.

30. Prove statement of Theorem : for all integers .

Assume the statement from Exercise 30 in section 2.1 that for all and in . Use this assumption and mathematical induction to prove that for all positive integers and arbitrary integers .

Let R be the set of all infinite sequences of real numbers, with the operations u+v=(u1,u2,u3,......)+(v1,v2,v3,......)=(u1+v1,u2+v2,u3+v3,.....) and cu=c(u1,u2,u3,......)=(cu1,cu2,cu3,......). Determine whether R is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.