Chapter 5.3, Problem 35ES

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.

3. Let f(z) = sin (22) + cos (T2) 2(22+1)(z+1) Compute f(z)dz over each of the contours/closed curves C1, C2, C3 and C4 shown below. Don't use any Al tool Don't send the same previous answer that was Al generated L 10 -c x show ur answer pe n and paper then take Send ur answer in pe n and paper don't rep uted ur self down

PLEASE SOLVE STEP BY STEP WITHOUT ARTIFICIAL INTELLIGENCE OR CHATGPT SOLVE BY HAND STEP BY STEP 4.- A beer at an unknown temperature is introduced into a refrigerator that has a constant temperature of 1°C. After 20 minutes, the temperature of the beer is 10°C, and after 40 minutes, the temperature of the beer is 6°C. a) Determine the temperature at which the beer was placed inside the refrigerator.b) How long will it take for the beer to reach 2°C?