Chapter 5.5, Problem 18ES

Prove let Aand B submodul of M A is large sub podule A large of B and B large of M. SM B Smale sub module B/A smal of M/A and As Mallof M. Give example and expleain caim. Amonorphism and split d) Determine the following group: Hom, (Q,Z) and Ho M₂ (Q, Q) and Hom (2/12, Q) =

Q2: Using the Laplace transform, find the solution for the following equation y"" +y" = 6et + 6t + 6. Suppose zero initial conditions (y"" (0) = y"(0) = y'(0) = y(0) = 0).

1- Let A = {A1, A2, ...), in which A, A, = 0, when i j. a) Is A a π-system? If not, which element(s) should be added to A to become a π-system? b) Prove that σ(A) consists of the finite or countable unions of elements of A; i.c., A E σ(A) if and only if there exists finite or countable sequence {n} such that A = U₁An (Hint: Let F be such class; prove that F is a σ-filed containing A.) c) Let p ≥ 0 be a sequence of non-negative real numbers with Σip₁ = 1. Using p₁'s, how do you construct a probability measure on σ(A)? (Hint: use extension theorem.) 2- Construct an example for which P(lim sup A,) = 1 and P(lim inf An) = 0.