Problem 1487
State the dominated convergence theorem for series. Difficulty: 3. |
Problem 1481
Apply Lebesgue's monotone convergence theorem to calculate \(\displaystyle \lim_{n\to\infty}\int_0^n \left(1+\frac xn\right)^n e^{-2x}\dx. \) Difficulty: 4. |
Problem 1482
True or false? If \(\displaystyle f_1\ge f_2\ge \dots\) are non-negative and Lebesgue-measurable then \(\displaystyle \lim \int f_n \,\mathrm{d}\lambda = \int (\lim f_n) \,\mathrm{d}\lambda? \) Difficulty: 4. |
Problem 1483
Let \(\displaystyle A=\{1,2\}\), and let \(\displaystyle \mu:A\to\R\) be the counting measure. State and explain Fatou's lemma in this situation. Difficulty: 4. |
Problem 1486
Derive the monotone convergence theorem from Fatou's lemma. Difficulty: 4. |
Problem 1493
Show using the Beppo Levi's theorem that if \(\displaystyle f_n\) is non-negative and \(\displaystyle \mu\)-measurable on a \(\displaystyle \mu\)-measurable set \(\displaystyle A\) and \(\displaystyle \int_A f_nd\mu<1/n^2\), then \(\displaystyle f_n\to 0\) \(\displaystyle \mu\)-a.e. Difficulty: 4. |
Problem 1485
Give a sequence \(\displaystyle f_n:[0,1]\to\R\) that converges pointwise, for which \(\displaystyle \lim\int_0^1f_n\) exists but \(\displaystyle \lim\int_0^1f_n\ne\int_0^1\lim f_n\). Difficulty: 5. |
Problem 1488
True or false? If \(\displaystyle f_n\) is non-negative and \(\displaystyle \mu\)-measurable on a \(\displaystyle \mu\)-measurable set \(\displaystyle A\) and \(\displaystyle \int_A f_nd\mu<1/n\) then \(\displaystyle f_n\to 0\) \(\displaystyle \mu\)-a.e. Difficulty: 5. |
Problem 1492
Show using the Borel-Cantelli lemma that if \(\displaystyle f_n\) is non-negative and \(\displaystyle \mu\)-measurable on a \(\displaystyle \mu\)-measurable set \(\displaystyle A\) and \(\displaystyle \int_A f_nd\mu<1/n^2\), then \(\displaystyle f_n\to 0\) \(\displaystyle \mu\)-a.e. Difficulty: 5. |
Problem 1480
True or false? If \(\displaystyle f_n:\R\to\R\) are Lebesgue-measurable, then it has a subsequence that converges a.e.? Difficulty: 8. |
Problem 1494
Show without Lebesgue theory that if \(\displaystyle f_n:[0,1]\to[0,1]\) is continuous for all \(\displaystyle n\) and \(\displaystyle f_n(x)\to 0\) for all \(\displaystyle x\in[0,1]\), then \(\displaystyle \int_0^1 f_n(x) \dx\to 0\) ! Difficulty: 8. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |