Question

If there exists a sequence $(\phi_n)$ of step functions such that $\phi_n \longrightarrow f$ almost everywhere on $[a,b]$, can we prove that $f\in L$?

score 107 · Answer 1 · 2026-04-15 19:51:28

107

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If there exists a sequence $(\phi_n)$ of step functions such that $\phi_n \longrightarrow f$ almost everywhere on $[a,b]$, can we prove that $f\in L$?

Published on 15 Apr 2026 - 19:51

score 314 · Answer 2 · 2026-04-14 23:42:10

314

Views

Help with a proof that $\mathop {\lim }\limits_{x \to + \infty } f\left( x \right) = 0$

Published on 14 Apr 2026 - 23:42

score 467 · Answer 3 · 2026-04-15 20:14:30

467

Views

If $\sup_n\int_E f_n(x)\ \mathsf dx\leq M\mu(E)$ then the measure of $\{x\in [0,\infty)\mid f(x)>M\}=0$.

Published on 15 Apr 2026 - 20:14

score 208 · Answer 4 · 2026-04-10 06:03:55

208

Views

Lebesgue Integral is Linear

Published on 10 Apr 2026 - 6:03

score 1.3k · Answer 5 · 2026-04-17 18:40:42

1.3k

Views

For what $p$ is $\frac{1}{(x(1+\ln(x)^2))^p}$ Lebesgue integrable?

Published on 17 Apr 2026 - 18:40

score 68 · Answer 6 · 2026-04-17 18:40:40

68

Views

Show that $\mu(f)\mu(1/f)\geq\mu(\Omega)^2$

Published on 17 Apr 2026 - 18:40

score 69 · Answer 7 · 2026-04-16 10:58:47

69

Views

Prove that the minimal set of functions for which these properties hold are step functions

Published on 16 Apr 2026 - 10:58

score 146 · Answer 8 · 2026-04-14 23:37:05

146

Views

Lebesgue integrability of $x^{-\frac{1}{2}}$

Published on 14 Apr 2026 - 23:37

score 74 · Answer 9 · 2026-04-12 10:11:49

74

Views

How is the line integration defined in the most general setting?

Published on 12 Apr 2026 - 10:11

score 133 · Answer 10 · 2026-04-16 19:03:01

133

Views

Find the derivative of $f(x) = \int_{-\infty}^\infty \frac{e^{-xy^2}}{1+y^2}\ dy.$

Published on 16 Apr 2026 - 19:03

score 98 · Answer 11 · 2026-04-17 14:23:44

98

Views

Calculate $\lim\limits_{n \to \infty} \int_0^1 \frac{n^p x^r\log x}{1 + n^2 x^2}dx$

Published on 17 Apr 2026 - 14:23

score 1k · Answer 12 · 2026-04-14 22:20:06

1k

Views

Multiplication operators on $L^2$

Published on 14 Apr 2026 - 22:20

score 87 · Answer 13 · 2026-04-17 18:40:39

87

Views

Conditions to ensure nice integrability of supremum of difference on neighborhood

Published on 17 Apr 2026 - 18:40

score 529 · Answer 14 · 2026-04-17 18:40:43

529

Views

$L^2(\mathbb{R})$ sequence such that $\sum_{n=1}^{\infty}\int_{\mathbb{R}}f_n(x)g(x)d\mu(x)=0$

Published on 17 Apr 2026 - 18:40

score 88 · Answer 15 · 2026-04-17 13:45:03

88

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prove that $\lim_{m \rightarrow \infty} \Sigma_{k=-m^2}^{m^2}|\int^{(k+1)/m}_{k/m}f(x)dx|=\|f\|_{L^1 (\Bbb R)}$.

Published on 17 Apr 2026 - 13:45

List Question

If there exists a sequence $(\phi_n)$ of step functions such that $\phi_n \longrightarrow f$ almost everywhere on $[a,b]$, can we prove that $f\in L$?

Help with a proof that $\mathop {\lim }\limits_{x \to + \infty } f\left( x \right) = 0$

If $\sup_n\int_E f_n(x)\ \mathsf dx\leq M\mu(E)$ then the measure of $\{x\in [0,\infty)\mid f(x)>M\}=0$.

Lebesgue Integral is Linear

For what $p$ is $\frac{1}{(x(1+\ln(x)^2))^p}$ Lebesgue integrable?

Show that $\mu(f)\mu(1/f)\geq\mu(\Omega)^2$

Prove that the minimal set of functions for which these properties hold are step functions

Lebesgue integrability of $x^{-\frac{1}{2}}$

How is the line integration defined in the most general setting?

Find the derivative of $f(x) = \int_{-\infty}^\infty \frac{e^{-xy^2}}{1+y^2}\ dy.$

Calculate $\lim\limits_{n \to \infty} \int_0^1 \frac{n^p x^r\log x}{1 + n^2 x^2}dx$

Multiplication operators on $L^2$

Conditions to ensure nice integrability of supremum of difference on neighborhood

$L^2(\mathbb{R})$ sequence such that $\sum_{n=1}^{\infty}\int_{\mathbb{R}}f_n(x)g(x)d\mu(x)=0$

prove that $\lim_{m \rightarrow \infty} \Sigma_{k=-m^2}^{m^2}|\int^{(k+1)/m}_{k/m}f(x)dx|=\|f\|_{L^1 (\Bbb R)}$.

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