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 ...continues
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 al ...continues
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 ...continues
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 ...continues
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 ...continues
Past analysis qualifying exam part 2
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 3 and 4. (See attached file for full problem description with ...continues
Past analysis qualifying exam part 3
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 5 and 6. (See attached file for full problem description with ...continues
Past analysis qualifying exam part 4
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 7 and 8. (See attached file for full problem description with ...continues
(See attached file for full problem description) I am using the book Methods of Real analysis by Richard Goldberg. --- Let and let D be a dense subset of E. If are continuous real-valued functions on E for n=1,2,…, and converges uniformly on D, prove that converges uniformly on E. ---
If the conjecture is true, prove it. If it's false, prove that its false by counterexample or a proof by contradiction.
2. Denote by -P the set of all negative integers, i.e., the set to which the number m belongs only in case there is a member n of P such that m = -n. If the number m is in -P and Z is a number such that m
