Question

Show that $E =\{(x,\alpha )\mid|f(x)|>\alpha\geq 0 \}$ is measurable.

score 109 · Answer 1 · 2026-04-16 15:00:44

109

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Show that $E =\{(x,\alpha )\mid|f(x)|>\alpha\geq 0 \}$ is measurable.

Published on 16 Apr 2026 - 15:00

score 182 · Answer 2 · 2026-04-13 15:01:10

182

Views

Lebesgue measure of some sets

Published on 13 Apr 2026 - 15:01

score 1k · Answer 3 · 2026-03-31 14:15:36

1k

Views

Compute Lebesgue measure of set of all real numbers in $[0,1]$ whose decimal representations don't contain the number 7

Published on 31 Mar 2026 - 14:15

score 638 · Answer 4 · 2026-04-14 09:57:29

638

Views

Show there is a set $H$ which is the intersection of countable many open sets containing $A$ with $\mu(H \setminus A)=0$

Published on 14 Apr 2026 - 9:57

score 63 · Answer 5 · 2026-04-10 06:06:29

63

Views

Positive measure set after projection

Published on 10 Apr 2026 - 6:06

score 66 · Answer 6 · 2026-04-30 08:00:59

66

Views

Show that $f$ is summable on $A$ and $\lim_{n\rightarrow \infty} \int_A f_n dm=\int_A f dm$

Published on 30 Apr 2026 - 8:00

score 82 · Answer 7 · 2026-04-30 08:07:16

82

Views

Use monotone convergence theorem to show that $f(x)=\frac{1}{x}$ is not summable on $(0,1)$

Published on 30 Apr 2026 - 8:07

score 521 · Answer 8 · 2026-04-15 07:37:15

521

Views

Product of $L^{1}$ Function and Exponentially Integrable Function

Published on 15 Apr 2026 - 7:37

score 163 · Answer 9 · 2026-04-17 02:51:12

163

Views

$f \in L^1 ((0,1))$, decreasing on $(0,1)$ implies $x f(x)\rightarrow 0$ as $x \rightarrow 0$

Published on 17 Apr 2026 - 2:51

score 110 · Answer 10 · 2026-04-30 07:56:53

110

Views

evaluating Lebesgue integral 3

Published on 30 Apr 2026 - 7:56

score 67 · Answer 11 · 2026-04-17 08:16:11

67

Views

show that $ \lim \limits _{n\to\infty} \frac {Vol(\{ x \in B \mid \lvert x_1 \rvert > \epsilon \}} {Vol(B)} \to 0$ where $B$ is the unit ball

Published on 17 Apr 2026 - 8:16

score 657 · Answer 12 · 2026-04-30 07:56:25

657

Views

An integrable function is finite a.e.

Published on 30 Apr 2026 - 7:56

score 96 · Answer 13 · 2026-04-30 09:14:00

96

Views

If $f_k\to f$ in measure, does $f_k\to f$?

Published on 30 Apr 2026 - 9:14

score 48 · Answer 14 · 2026-04-30 09:13:12

48

Views

Show that $f(x)=\int_{[a,x]} g \, dm$ is continuous on $[a,b]$

Published on 30 Apr 2026 - 9:13

score 78 · Answer 15 · 2026-04-30 09:10:00

78

Views

Riesz-Fischer Theorem with Variable Exponent

Published on 30 Apr 2026 - 9:10

List Question

Show that $E =\{(x,\alpha )\mid|f(x)|>\alpha\geq 0 \}$ is measurable.

Lebesgue measure of some sets

Compute Lebesgue measure of set of all real numbers in $[0,1]$ whose decimal representations don't contain the number 7

Show there is a set $H$ which is the intersection of countable many open sets containing $A$ with $\mu(H \setminus A)=0$

Positive measure set after projection

Show that $f$ is summable on $A$ and $\lim_{n\rightarrow \infty} \int_A f_n dm=\int_A f dm$

Use monotone convergence theorem to show that $f(x)=\frac{1}{x}$ is not summable on $(0,1)$

Product of $L^{1}$ Function and Exponentially Integrable Function

$f \in L^1 ((0,1))$, decreasing on $(0,1)$ implies $x f(x)\rightarrow 0$ as $x \rightarrow 0$

evaluating Lebesgue integral 3

show that $ \lim \limits _{n\to\infty} \frac {Vol(\{ x \in B \mid \lvert x_1 \rvert > \epsilon \}} {Vol(B)} \to 0$ where $B$ is the unit ball

An integrable function is finite a.e.

If $f_k\to f$ in measure, does $f_k\to f$?

Show that $f(x)=\int_{[a,x]} g \, dm$ is continuous on $[a,b]$

Riesz-Fischer Theorem with Variable Exponent

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