Looking for Problem

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):

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

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

 

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.

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

 

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

[tex] A = \{ x \in X \mid x \in A_n [/tex] for all but finitely many [tex] n \geq 1 \}[/tex],

[tex] B = \{ x \in X \mid x \in A_n [/tex] for infinitely many [tex] n \geq 1\}[/tex].

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

(a) [tex] \bigcup_{k=1}^{\infty} \bigcap_{n=k}^{\infty} A_n [/tex]

(b) [tex] \bigcap_{k=1}^{\infty} \bigcup_{n=k}^{\infty} A_n [/tex]


Question 2: Further suppose that [tex] A_1, A_2, ... \in M [/tex] and

[tex] \sum_{n=1}^{\infty} \mu(A_n) < \infty [/tex]

Prove that [tex] \mu(B) = 0 [/tex].

 

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

 

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.

 

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