Showing a quotient space is a complete metric space; Finite measurable space; Symmetric difference; Equivalence relations

Let (X,B,mu) be a complete, finite measuable space. For each C,D in B, set d(C,D) = mu (C / D) where C / D is the symmetric difference of C and D. We say that two measurable sets C,D are equivalent if d(C,D)=0 (this is an equivalence relation). Let E be the set of equivalence classes, and show that d introduces a metric on E and that (E,d) is a complete metric space.

Almost every point is a density point

A point x of a measurable subset A of the reals is called a density point if m( A intersection [x-h, x+h] ) / 2h goes to 1 as h goes to 0 where m is the Lebesgue measure. Prove that if A is a set of positive, finite Lebesgue measure, then almost every point of A is a density point. I would like to note that I can use any results from Royden's Real Analysis book.

Properties of additive functions; Bounded; Continuous; Measurable

Let f : R --> R be an additive function i.e. f(x+y) = f(x) + f(y) for all x,y in R. 1. If f is bounded at a point, then f is continuous at that point. 2. If f is measurable, then f is linear i.e. f(x) = cx for some c in R. I have already proved that f is continuous if and only if f is linear, and I have proven that if f is continuous at a point then it is continuous everywhere.

Space of functions is sequentially compact

Let C_0 be the space of functions f:R --> R such that lim f(x) = 0 as x goes to infinity and negative infinity C_0 becomes a metric space with sup-norm ||f|| = sup { |f(x)| : x in R } Prove that if A is a family of functions in C_0 such that A is uniformly bounded and equicontinuous, then every sequence of functions from A has a convergent subsequence in the sup-norm. My current work is to try to reformulate this into a problem where I can use Ascoli-Azrela theorem. One idea was to regard C_0 as the space of functions on the circle as I have seen a similar construction in topology. However, I am unable to fill in the details.

Lebesque measurable sets in R^n.

Prove that lebesque measurable sets in R^n form sigma algebra. ( Please use basic definition when you talk about the lebesgue measurable sets in R^n). The def we have is: (k_1)^(m)={ -1/2 + m_i =< x_i =< 1/2+ m} m=(m_1,m_2,...,m_n) m belongs to z^d Now we say that A in R^n is Lebesque measurable set in R^n if A intersection (k_1)^(m) is lebesque measurable for every m in z^d.