- #1

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- Thread starter Dragonfall
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- #1

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- #2

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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).

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).

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- #3

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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]

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

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- #4

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

- #5

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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].

[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].

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- #6

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

- #7

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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.

- #8

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

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