Problem 1457
Let \(\displaystyle f:\R\to\R\) be monotonically increasing and for all \(\displaystyle a\le b\) let \(\displaystyle \mu\big([a,b]\big)=f(b+0)-f(a-0)\). What measure does this generate? Difficulty: 4. |
Problem 1451
Let \(\displaystyle \mu\) be a translation-invariant measure on the Borel-sets of \(\displaystyle \R\), for which \(\displaystyle \mu\big([0,1]\big)<\infty\). Show that \(\displaystyle \mu\) is the Lebesgue-measure up to a constant multiple. Difficulty: 5. |
Problem 1453
Show that if \(\displaystyle H\subset\R\) satisfies \(\displaystyle \displaystyle \olambda((a,b)\cap H)<\frac{99}{100}(b-a)\) for all \(\displaystyle a<b\), then \(\displaystyle H\) is a null-set. Difficulty: 5. |
Problem 1460
Let \(\displaystyle f:\R\to\R\) be monotonically increasing and \(\displaystyle \mu_f\) the Lebesgue-Stieltjes measure generated by \(\displaystyle f\). Show that for any Borel-set \(\displaystyle H\) there are \(\displaystyle F_{\sigma}\) \(\displaystyle B\subset H\) and \(\displaystyle G_\delta\) \(\displaystyle K\supset H\) sets for which \(\displaystyle \mu_f(B)=\mu_f(K)=\mu_f(H)\). Difficulty: 5. |
Problem 1443
For any \(\displaystyle \varepsilon >0\) give \(\displaystyle G\subset\R\) which is open and dense and for which \(\displaystyle \olambda(G)<\varepsilon\). Difficulty: 8. |
Problem 1445
Construct a Borel-set \(\displaystyle H\subset\R\) for which \(\displaystyle {\lambda((a,b)\cap H)>0}\) and \(\displaystyle {\lambda((a,b)\setminus H)>0}\) for any \(\displaystyle a<b\). Difficulty: 8. |
Problem 1462
(a) Show that if \(\displaystyle A\subset\R^p\) is measurable and \(\displaystyle \lambda(A)>0\), then \(\displaystyle A-A\) contains a ball centered at the origin. (Steinhaus) (b) Show that if \(\displaystyle A, B\subset\R^p\) are measurable with positive measure, then \(\displaystyle A+B\) has a nonempty interior. (c) Show that if \(\displaystyle A\subset\R^p\) measurable with positive measure and \(\displaystyle B\subset\R^p\) has positive outer measure, then \(\displaystyle A+B\) has a nonempty interior. Difficulty: 8. |
Problem 1454
Can one find continuum many Lebesgue measurable sets in \(\displaystyle [0,1]\) all of measure \(\displaystyle 1/2\) such that for any two the intersection has measure \(\displaystyle 1/4\)? Difficulty: 9. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |