Problem 1426
Let \(\displaystyle \mathcal{A}\) and \(\displaystyle \mathcal{B}\) be \(\displaystyle \sigma\)-rings. Describe the \(\displaystyle \sigma\)-ring generated by \(\displaystyle \mathcal{A}\cup\mathcal{B}\). Difficulty: 3. |
Problem 1429
(a) What ring do the half-lines \(\displaystyle [a,\infty)\) generate? (b) What \(\displaystyle \sigma\)-ring do the half-lines \(\displaystyle [a,\infty)\) generate? (c) What is the smallest cardinality of a generating set of the \(\displaystyle \sigma\)-ring of Borel-sets? Difficulty: 3. |
Problem 1428
Let \(\displaystyle \mathcal{T}\) be the collection of the sets \(\displaystyle [a,b)\times[c,d)\). (a) Show that \(\displaystyle \mathcal{T}\) is a semi-ring. (b) What ring does \(\displaystyle \mathcal{T}\) generate? (c) Show that \(\displaystyle f:\mathcal{T}\to\R\) is additive if and only if there is \(\displaystyle g:\R^2\to\R\) for which \(\displaystyle f\big([a,b)\times[c,d)\big)=g(b,d)-g(a,d)-g(b,c)+g(a,b)\). Difficulty: 5. |
Problem 1430
Show that all open sets are \(\displaystyle F_\sigma\), and all closed sets are \(\displaystyle G_\delta\). Difficulty: 5. |
Problem 1434
Show that \(\displaystyle F_{\sigma\delta\sigma\delta}(\R^n)\subset G_{\delta\sigma\delta\sigma\delta}(\R^n)\). Difficulty: 5. |
Problem 1432
Prove that sets with property \(\displaystyle F_\sigma\), respectively \(\displaystyle G_\delta\), are closed to finite union and intersection. Difficulty: 6. |
Problem 1427
What it is the smallest possible cardinality of an infinite \(\displaystyle \sigma\)-ring? Difficulty: 7. Answer (final result) is provided for this problem. |
Problem 1431
Prove that if \(\displaystyle f:\R \to \R\), then the set of points of continuity is Borel, and give as small as possible of Borel-class (e.g.. \(\displaystyle G_{\delta\sigma\delta\sigma\delta\sigma\delta\sigma}\)), to which it still belongs. Difficulty: 7. Solution is available for this problem. |
Problem 1436
Let \(\displaystyle f_n :[a,b]\to \R\) be continuous for all \(\displaystyle n\). Prove that \(\displaystyle \{ x: f_n (x)\,\text{convergent}\}\) is a Borel-set, and give a Borel-class as small as possible to which it still belongs. Difficulty: 7. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |