Sets of null (Lebesgue)-measure and sigma compacts
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Let $Esubset{mathbb R}$ be a set such that $m(E)=0$, that is, with zero Lebesgue measure.
¿It is possible to find $Fsubset{mathbb R}$ such that $Esubset F$, $m(F)=0$ and $F$ is $sigma$-compact (that is, numerable union of compact sets)?
Obviously if $E$ is numerable, this is trivial.
Moreover, if $m(overline{E})=0$ (where $overline{E}$ is the clousure of $E$), it is also true. Just take $F=overline{E}$ and $overline{E}=bigcup_{ninmathbb N} overline{E}cap[-n,n]$.
IS there a null-set $E$ such that $m(overline{E})>0$ but $E$ cannot be covered by a null-sigma compact set?
I am also interested when $Esubsetmathbb T$, where $mathbb T$ is the unit circle.
measure-theory lebesgue-measure compactness
asked Nov 16 at 10:41
Tito Eliatron
1,012621
add a comment |
up vote
1
down vote
favorite
Let $Esubset{mathbb R}$ be a set such that $m(E)=0$, that is, with zero Lebesgue measure.
¿It is possible to find $Fsubset{mathbb R}$ such that $Esubset F$, $m(F)=0$ and $F$ is $sigma$-compact (that is, numerable union of compact sets)?
Obviously if $E$ is numerable, this is trivial.
Moreover, if $m(overline{E})=0$ (where $overline{E}$ is the clousure of $E$), it is also true. Just take $F=overline{E}$ and $overline{E}=bigcup_{ninmathbb N} overline{E}cap[-n,n]$.
IS there a null-set $E$ such that $m(overline{E})>0$ but $E$ cannot be covered by a null-sigma compact set?
I am also interested when $Esubsetmathbb T$, where $mathbb T$ is the unit circle.
measure-theory lebesgue-measure compactness
asked Nov 16 at 10:41
Tito Eliatron
1,012621
The real question is if there existe a non numerable nulo set whose clausure has positive mensure and such that it is not contained on any null sigma compact set.
– Tito Eliatron
Nov 18 at 11:02
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $Esubset{mathbb R}$ be a set such that $m(E)=0$, that is, with zero Lebesgue measure.
¿It is possible to find $Fsubset{mathbb R}$ such that $Esubset F$, $m(F)=0$ and $F$ is $sigma$-compact (that is, numerable union of compact sets)?
Obviously if $E$ is numerable, this is trivial.
Moreover, if $m(overline{E})=0$ (where $overline{E}$ is the clousure of $E$), it is also true. Just take $F=overline{E}$ and $overline{E}=bigcup_{ninmathbb N} overline{E}cap[-n,n]$.
IS there a null-set $E$ such that $m(overline{E})>0$ but $E$ cannot be covered by a null-sigma compact set?
I am also interested when $Esubsetmathbb T$, where $mathbb T$ is the unit circle.
measure-theory lebesgue-measure compactness
asked Nov 16 at 10:41
Tito Eliatron
1,012621
Let $Esubset{mathbb R}$ be a set such that $m(E)=0$, that is, with zero Lebesgue measure.
¿It is possible to find $Fsubset{mathbb R}$ such that $Esubset F$, $m(F)=0$ and $F$ is $sigma$-compact (that is, numerable union of compact sets)?
Obviously if $E$ is numerable, this is trivial.
Moreover, if $m(overline{E})=0$ (where $overline{E}$ is the clousure of $E$), it is also true. Just take $F=overline{E}$ and $overline{E}=bigcup_{ninmathbb N} overline{E}cap[-n,n]$.
IS there a null-set $E$ such that $m(overline{E})>0$ but $E$ cannot be covered by a null-sigma compact set?
I am also interested when $Esubsetmathbb T$, where $mathbb T$ is the unit circle.
measure-theory lebesgue-measure compactness
measure-theory lebesgue-measure compactness
asked Nov 16 at 10:41
Tito Eliatron
1,012621
asked Nov 16 at 10:41
Tito Eliatron
1,012621
edited yesterday
asked Nov 16 at 10:41
Tito Eliatron
1,012621
asked Nov 16 at 10:41
Tito Eliatron
1,012621
asked Nov 16 at 10:41
Tito Eliatron
1,012621
1,012621
The real question is if there existe a non numerable nulo set whose clausure has positive mensure and such that it is not contained on any null sigma compact set.
– Tito Eliatron
Nov 18 at 11:02
add a comment |
The real question is if there existe a non numerable nulo set whose clausure has positive mensure and such that it is not contained on any null sigma compact set.
– Tito Eliatron
Nov 18 at 11:02
The real question is if there existe a non numerable nulo set whose clausure has positive mensure and such that it is not contained on any null sigma compact set.
– Tito Eliatron
Nov 18 at 11:02
The real question is if there existe a non numerable nulo set whose clausure has positive mensure and such that it is not contained on any null sigma compact set.
– Tito Eliatron
Nov 18 at 11:02
add a comment |
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The real question is if there existe a non numerable nulo set whose clausure has positive mensure and such that it is not contained on any null sigma compact set.
– Tito Eliatron
Nov 18 at 11:02