Analysis Problem: Find Solution in 1 Week

[Dragonfall]

I'm looking for a good analysis problem (solved or otherwise) that will keep me busy for, say, a week. Anything up to Lebesgue integration is ok.

 

[BSMSMSTMSPHD]

It's not a very interesting problem, but it will require a bit of paper.

Consider the following 11 subsets of the real numbers (let a and b be any real number):

$$E_1 = \{(a,b)\}$$
$$E_2 = \{[a,b)\}$$
$$E_3 = \{(a,b]\}$$
$$E_4 = \{[a,b]\}$$
$$E_5 = \{(a,\infty)\}$$
$$E_6 = \{[a,\infty)\}$$
$$E_7 = \{(-\infty,b)\}$$
$$E_8 = \{(-\infty,b]\}$$
$$E_9 = \{all open sets\}$$
$$E_1_0 = \{all closed sets\}$$
$$E_1_1 = \{all compact sets\}$$

Show that the sigma-algebras generated by each of these sets is the same (that is, that they are all subsets of each other).

 

[BSMSMSTMSPHD]

Here's another one I like. Technically it's a calculus question, but you can make it Analysis by justifying each of the steps involved. You'll need at least one major theorem.

$$\lim_{n \rightarrow \infty} \int_{0}^{\infty} e^{-2x} \sum_{k=0}^{n} \frac{x^k}{k!} dx$$

 

[Dragonfall]

Thanks, but I was thinking something a little more theoretical and less repetitive.

 

[BSMSMSTMSPHD]

Suppose $$(X, M, \mu)$$ is a measure space. Suppose $$A_1, A_2,$$... are subsets of X. Let:

$$A = \{ x \in X \mid x \in A_n$$ for all but finitely many $$n \geq 1 \}$$,

$$B = \{ x \in X \mid x \in A_n$$ for infinitely many $$n \geq 1\}$$.

Question 1: Which, if any, of the following are equal to A or B?

(a) $$\bigcup_{k=1}^{\infty} \bigcap_{n=k}^{\infty} A_n$$

(b) $$\bigcap_{k=1}^{\infty} \bigcup_{n=k}^{\infty} A_n$$Question 2: Further suppose that $$A_1, A_2, ... \in M$$ and

$$\sum_{n=1}^{\infty} \mu(A_n) < \infty$$

Prove that $$\mu(B) = 0$$.

 

[BSMSMSTMSPHD]

Give an example of a ring of subsets of a set X that is not a sigma-ring.

 

[ircdan]

Suppose f:[a,b]->R is bounded and continuous except a finite number of points z_1, ..., z_k. Prove f:[a,b]->R is riemann integrable. (I believe this is also true if f is continuous except at a countable number of points)

I'm not sure my proof is correct but I'd love to see yours if you come up with one so we can compare.

 

[DeadWolfe]

Actually, it is true exactly when the discontinuities form a set of measure zero.