Question

Show that $ \lim_{k \to \infty} k^n \int_{(0,k^{-1})^n} f(x)dx=f(0)$

score 80 · Answer 1 · 2023-08-29 04:47:38

80

Views

Show that $ \lim_{k \to \infty} k^n \int_{(0,k^{-1})^n} f(x)dx=f(0)$

Published on 29 Aug 2023 - 4:47

score 52 · Answer 2 · 2026-04-12 03:16:56

52

Views

Integral with respect to a restricted measure

Published on 12 Apr 2026 - 3:16

score 89 · Answer 3 · 2023-08-29 16:41:16

89

Views

Proving $\frac{1}{\epsilon}\int_{-\infty}^\infty\frac{f(x)}{\frac{1}{\epsilon^2}+(x-\frac{1}{\epsilon})^2}\,dx=\pi f(\frac{1}{\epsilon})+O(\epsilon).$

Published on 29 Aug 2023 - 16:41

score 103 · Answer 4 · 2026-04-13 00:15:58

103

Views

Please check my proof that a weird-integral-like $R^2 \to R$ function is continuous

Published on 13 Apr 2026 - 0:15

score 138 · Answer 5 · 2026-03-25 20:22:57

138

Views

Understanding proof of "if $f$ is non-negative, then it is the supremum of simple functions."

Published on 25 Mar 2026 - 20:22

score 81 · Answer 6 · 2023-09-07 13:39:11

81

Views

Lebesgue integral of gradient

Published on 07 Sep 2023 - 13:39

score 46 · Answer 7 · 2026-04-13 04:38:56

46

Views

Why not expand the Lebesgue integral by summing up a series of multiple Lebesgue integrals?

Published on 13 Apr 2026 - 4:38

score 46 · Answer 8 · 2026-03-25 08:12:51

46

Views

On the set of points where a Henstock-Kurzweil integrable function fails to be Lebesgue integrable

Published on 25 Mar 2026 - 8:12

score 82 · Answer 9 · 2026-04-11 20:21:17

82

Views

Proof of Strassen's theorem in $\mathbb{R}^2$

Published on 11 Apr 2026 - 20:21

score 34 · Answer 10 · 2026-04-12 08:59:59

34

Views

Lebesgue Integration in generalized polar coordinates of a compact smooth hypersurface

Published on 12 Apr 2026 - 8:59

score 80 · Answer 11 · 2026-03-30 05:00:05

80

Views

Different definitions of the canonical form of simple functions (issue with constant reals different from $0$)

Published on 30 Mar 2026 - 5:00

score 47 · Answer 12 · 2023-09-15 16:08:19

47

Views

$\| f \|_{L^1} > 0 $ is stronger than simply $f \neq 0$. But how much?

Published on 15 Sep 2023 - 16:08

score 39 · Answer 13 · 2026-04-11 21:18:14

39

Views

How to prove that limit exists and equals specific value/function? (Durrett Problem 1.5.7)

Published on 11 Apr 2026 - 21:18

score 36 · Answer 14 · 2026-04-10 22:22:27

36

Views

Connection between Abstract Integral and Riemann Integral

Published on 10 Apr 2026 - 22:22

score 84 · Answer 15 · 2026-04-13 03:22:59

84

Views

Well definess of the Lebesgue integral

Published on 13 Apr 2026 - 3:22

List Question

Show that $ \lim_{k \to \infty} k^n \int_{(0,k^{-1})^n} f(x)dx=f(0)$

Integral with respect to a restricted measure

Proving $\frac{1}{\epsilon}\int_{-\infty}^\infty\frac{f(x)}{\frac{1}{\epsilon^2}+(x-\frac{1}{\epsilon})^2}\,dx=\pi f(\frac{1}{\epsilon})+O(\epsilon).$

Please check my proof that a weird-integral-like $R^2 \to R$ function is continuous

Understanding proof of "if $f$ is non-negative, then it is the supremum of simple functions."

Lebesgue integral of gradient

Why not expand the Lebesgue integral by summing up a series of multiple Lebesgue integrals?

On the set of points where a Henstock-Kurzweil integrable function fails to be Lebesgue integrable

Proof of Strassen's theorem in $\mathbb{R}^2$

Lebesgue Integration in generalized polar coordinates of a compact smooth hypersurface

Different definitions of the canonical form of simple functions (issue with constant reals different from $0$)

$\| f \|_{L^1} > 0 $ is stronger than simply $f \neq 0$. But how much?

How to prove that limit exists and equals specific value/function? (Durrett Problem 1.5.7)

Connection between Abstract Integral and Riemann Integral

Well definess of the Lebesgue integral

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