Question

Image of Cantor set under Cantor-Lebesgue function

score 1k · Answer 1 · 2014-07-19 01:02:04

1k

Views

Image of Cantor set under Cantor-Lebesgue function

Published on 19 Jul 2014 - 1:02

score 74 · Answer 2 · 2026-04-01 08:06:47

74

Views

Let $f_n:\mathbb{R}\rightarrow [0, 1]$ be functions such that $\sup_{x \in \mathbb{R}}f_n(x) = 1/n$ and $||f||_1 = 1$.

Published on 01 Apr 2026 - 8:06

score 1.2k · Answer 3 · 2026-04-01 08:06:13

1.2k

Views

Find a non-negative function on [0,1] such that $t\cdot m(\{x:f(x) \geq t\}) \to 0$ that is not Lebesgue Integrable

Published on 01 Apr 2026 - 8:06

score 113 · Answer 4 · 2026-04-01 09:46:36

113

Views

For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$.

Published on 01 Apr 2026 - 9:46

score 519 · Answer 5 · 2014-07-21 16:20:22

519

Views

Reverse Fatou's lemma on probability space

Published on 21 Jul 2014 - 16:20

score 62 · Answer 6 · 2014-07-21 16:22:02

62

Views

Can we identify Fourier transform of continuous compacltly supported functions with finte complex Borel measure?

Published on 21 Jul 2014 - 16:22

score 68 · Answer 7 · 2026-04-01 09:46:26

68

Views

Show that $\int_E f (x, y) dx$ is differentiable with respect to $y$ and $\frac{d}{dy}\int_E f(x,y)dx=\int_E \frac{d}{dy}f(x,y)dx.$

Published on 01 Apr 2026 - 9:46

score 336 · Answer 8 · 2026-04-01 09:46:36

336

Views

Let $(X,μ)$ be a measure space. Find a necessary and sufficient condition on $(X,μ)$ that $L_q(E) ⊂ L_p(E)$ for all $1 ≤ p < q ≤ ∞.$

Published on 01 Apr 2026 - 9:46

score 2.7k · Answer 9 · 2014-07-22 01:01:31

2.7k

Views

Product of Lebesgue and counting measures

Published on 22 Jul 2014 - 1:01

score 102 · Answer 10 · 2014-07-22 14:36:06

102

Views

Without Lebesgue

Published on 22 Jul 2014 - 14:36

score 185 · Answer 11 · 2026-04-01 09:46:36

185

Views

Find $\lim_{n \rightarrow \infty}$$\int_0^n(1 + \frac{-x}{n})^n\cos(\frac{x}{\sqrt{n}})e^{x/2}dx$

Published on 01 Apr 2026 - 9:46

score 537 · Answer 12 · 2026-04-01 09:50:42

537

Views

Lebesgue integral of $\chi_{\mathbb{Q}}: \mathbb{R} \rightarrow \mathbb{R}$

Published on 01 Apr 2026 - 9:50

score 905 · Answer 13 · 2014-07-26 17:23:57

905

Views

Absolute continuity of pushforward measure

Published on 26 Jul 2014 - 17:23

score 6.3k · Answer 14 · 2014-07-26 19:31:19

6.3k

Views

Finite additivity in outer measure

Published on 26 Jul 2014 - 19:31

score 425 · Answer 15 · 2026-04-01 11:04:46

425

Views

Show that $\int_{\mathbb{R}^n}f_1f_2 ...f_n dx_1 ...dx_n ≤ (I_1 ...I_n)^{1/(n−1)}.$

Published on 01 Apr 2026 - 11:04

List Question

Image of Cantor set under Cantor-Lebesgue function

Let $f_n:\mathbb{R}\rightarrow [0, 1]$ be functions such that $\sup_{x \in \mathbb{R}}f_n(x) = 1/n$ and $||f||_1 = 1$.

Find a non-negative function on [0,1] such that $t\cdot m(\{x:f(x) \geq t\}) \to 0$ that is not Lebesgue Integrable

For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$.

Reverse Fatou's lemma on probability space

Can we identify Fourier transform of continuous compacltly supported functions with finte complex Borel measure?

Show that $\int_E f (x, y) dx$ is differentiable with respect to $y$ and $\frac{d}{dy}\int_E f(x,y)dx=\int_E \frac{d}{dy}f(x,y)dx.$

Let $(X,μ)$ be a measure space. Find a necessary and sufficient condition on $(X,μ)$ that $L_q(E) ⊂ L_p(E)$ for all $1 ≤ p < q ≤ ∞.$

Product of Lebesgue and counting measures

Without Lebesgue

Find $\lim_{n \rightarrow \infty}$$\int_0^n(1 + \frac{-x}{n})^n\cos(\frac{x}{\sqrt{n}})e^{x/2}dx$

Lebesgue integral of $\chi_{\mathbb{Q}}: \mathbb{R} \rightarrow \mathbb{R}$

Absolute continuity of pushforward measure

Finite additivity in outer measure

Show that $\int_{\mathbb{R}^n}f_1f_2 ...f_n dx_1 ...dx_n ≤ (I_1 ...I_n)^{1/(n−1)}.$

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