Difference Between Measurable Functions.

I have been told that both 1 and 2 are true, but I don't understand why. Since the set {x: f(x)=g(x)} is null, it seems to me that d(x) should be only be zero on the null set, whereas apparently it is zero everywhere EXCEPT the null set. It may be a typo, but more likely I'm just not seeing something incredibly obvious. Also, I'm told that the truth of 1 implies 2, but again I don't see why.

Determine whether the set of cylinders of the given pair of measure spaces is contained in (is a subset of) the set of rectangles of that pair of measure spaces.

Let (Omega_1, F_1, P_1) and (Omega_2, F_2, P_2) be the following measure spaces: Omega_1 = {a, b}, F_1 is the sigma algebra of all subsets of Omega_1, and P_1 is a measure on Omega_1. Omega_2 = {c, d}, F_2 is the sigma algebra of all subsets of Omega_2, and P_2 is a measure on Omega_2. Determine the makeup of the set C of cylinders of Omega_1 and the set R of rectangles of Omega_2, and show that C is a subset of R.

Convergence and Uniform Convergence

see attached.

Retractions

Let A_0 be contained in A_1 contained in A_2 and so on be a nested sequence of subspaces of X such that the union of all A_n is X and such that An contained in the interior of A_(n+1). Suppose for each n, there is a retraction r_n:A_(n+1) to An. Prove there is a retraction r: X to A_0.