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

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 believe I am close to solving these simple problems, but they just aren't coming out. 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.

Past analysis qualifying exam part 1

I plan to take the analysis qualifying exam next time it is offered and I would like detailed solutions to a previous exam (some of the problems are fairly basic, but I would like to see solutions to all eight problems to be thorough). Note: This is for problems 1 and 2. (See attached file for full problem description with proper symbols) --- Analysis Qualifying Exam Spring 2004 1. Let A and B be two nonempty sets of real numbers. Define A+B = {a+b: a A and b B}. (a) Show that if A is open, then A+B is open. (b) If A and B are both closed, is A+B closed? Justify your answer. 2. Let be differentiable for and A as . Prove that there is a sequence such that A. Gi... click for more