Problem 1463
Prove that if \(\displaystyle f:\R\to\R\) is monotonic, then it is Borel-measurable. Difficulty: 2. |
Problem 1464
Prove that the composition of Borel-measurable functions is Borel-measurable. Difficulty: 2. |
Problem 1470
Let \(\displaystyle f:\R\to\R\) be Borel-measurable, and \(\displaystyle g:M\to\R\) measurable for some \(\displaystyle (M,\mu)\) measure space. Prove that \(\displaystyle f\circ g\) is \(\displaystyle \mu\)-measurable. Difficulty: 2. |
Problem 1471
True or false? If \(\displaystyle f:[a,b]\to\R\) is Riemann-integrable, then it is Borel-measurable? Difficulty: 2. |
Problem 1472
Let \(\displaystyle A\subset\R\) be Lebesgue-measurable and \(\displaystyle \chi_A(x)=\begin{cases}1&x\in A\\ 0 & x\not\in A\end{cases}\). Show that \(\displaystyle \int_{\R}\chi_A \,\mathrm{d}\lambda=\lambda(A)\). Difficulty: 2. |
Problem 1466
Show that if \(\displaystyle f:[a,b]\to\R\) is Lebesgue-measurable, then there is \(\displaystyle g:[a,b]\to\R\) Borel-measurable such that \(\displaystyle f=g\) a.e. Difficulty: 4. |
Problem 1473
Show that if \(\displaystyle f>0\) on a \(\displaystyle \mu\)-measurable \(\displaystyle A\) such that \(\displaystyle \mu(A)>0\), then \(\displaystyle \int_A f \,\mathrm{d}\mu > 0\). Difficulty: 5. |
Problem 1478
Is there any measurable function \(\displaystyle f:\R\to [0,\infty)\), whose integral over any interval is \(\displaystyle +\infty\)? Difficulty: 5. |
Problem 1477
True or false? If \(\displaystyle f[a,b]\to\R\) is bounded and Lebesgue-integrable, then there is a \(\displaystyle g:[a,b]\to\R\) that is Riemann-integrable and for which \(\displaystyle f=g\) a.e.? Difficulty: 7. |
Problem 1468
Construct a function \(\displaystyle f:[0,1]\to\R\) whose restriction to any set with full measure is not continuous. Difficulty: 9. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |