Question on strictly positive measure of a level set












2












$begingroup$


Let $f in W^{2,p}(U)$ for $1 le p lt infty$ where $U$ is a compact subset of $mathbb R^2$ and $g in L^{infty}(U)$ with $g(f)=f$ almost everywhere on $U$. I have trouble understanding why the following holds:




Assume that the set ${f=0}$ is non empty and moreover that $g lt 1$ in ${f=0}$. Then it has a strictly
positive lebesgue measure.




I know that ${f=0}$ is a (relatively) closed set but that is not enough for the strict inequality. What do I miss? Is it because it contains an open set?



Any help/hints are much appreciated! Thanks in advance!






real-analysis analysis measure-theory lebesgue-measure







share|cite|improve this question











$endgroup$












  • $begingroup$
    Do you not have any additional assumptions? Unless I'm missing something, $f(x) = x_1^2+x_2^2$ is an easy counterexample.
    $endgroup$
    – ktoi
    Nov 28 '18 at 21:26












  • $begingroup$
    @ktoi I just edited my question...
    $endgroup$
    – kaithkolesidou
    Dec 3 '18 at 11:16






  • 1




    $begingroup$
    What is the connection between $f$ and $g$. The example given by ktoi is still valid with $g=0$.
    $endgroup$
    – Kavi Rama Murthy
    Dec 3 '18 at 12:01










  • $begingroup$
    @KaviRamaMurthy I'm sorry I didn't mention it from the beginning. I edited
    $endgroup$
    – kaithkolesidou
    Dec 3 '18 at 12:16










  • $begingroup$
    @kaithkolesidou What do you mean by $g(f)$? If it's functional composition, then the assumption $g<1$ on ${f = 0}$ doesn't really tell you anything and taking $g(x) = x$ my example still remains valid. Also it seems like you need more regularity assumptions on $g$; if ${ f = 0 }$ is a null set then the condition $g < 1$ on ${f = 0}$ gives hardly any restriction on $g.$
    $endgroup$
    – ktoi
    Dec 4 '18 at 19:07
















2












$begingroup$


Let $f in W^{2,p}(U)$ for $1 le p lt infty$ where $U$ is a compact subset of $mathbb R^2$ and $g in L^{infty}(U)$ with $g(f)=f$ almost everywhere on $U$. I have trouble understanding why the following holds:




Assume that the set ${f=0}$ is non empty and moreover that $g lt 1$ in ${f=0}$. Then it has a strictly
positive lebesgue measure.




I know that ${f=0}$ is a (relatively) closed set but that is not enough for the strict inequality. What do I miss? Is it because it contains an open set?



Any help/hints are much appreciated! Thanks in advance!






real-analysis analysis measure-theory lebesgue-measure







share|cite|improve this question











$endgroup$












  • $begingroup$
    Do you not have any additional assumptions? Unless I'm missing something, $f(x) = x_1^2+x_2^2$ is an easy counterexample.
    $endgroup$
    – ktoi
    Nov 28 '18 at 21:26












  • $begingroup$
    @ktoi I just edited my question...
    $endgroup$
    – kaithkolesidou
    Dec 3 '18 at 11:16






  • 1




    $begingroup$
    What is the connection between $f$ and $g$. The example given by ktoi is still valid with $g=0$.
    $endgroup$
    – Kavi Rama Murthy
    Dec 3 '18 at 12:01










  • $begingroup$
    @KaviRamaMurthy I'm sorry I didn't mention it from the beginning. I edited
    $endgroup$
    – kaithkolesidou
    Dec 3 '18 at 12:16










  • $begingroup$
    @kaithkolesidou What do you mean by $g(f)$? If it's functional composition, then the assumption $g<1$ on ${f = 0}$ doesn't really tell you anything and taking $g(x) = x$ my example still remains valid. Also it seems like you need more regularity assumptions on $g$; if ${ f = 0 }$ is a null set then the condition $g < 1$ on ${f = 0}$ gives hardly any restriction on $g.$
    $endgroup$
    – ktoi
    Dec 4 '18 at 19:07














2












2








2


2



$begingroup$


Let $f in W^{2,p}(U)$ for $1 le p lt infty$ where $U$ is a compact subset of $mathbb R^2$ and $g in L^{infty}(U)$ with $g(f)=f$ almost everywhere on $U$. I have trouble understanding why the following holds:




Assume that the set ${f=0}$ is non empty and moreover that $g lt 1$ in ${f=0}$. Then it has a strictly
positive lebesgue measure.




I know that ${f=0}$ is a (relatively) closed set but that is not enough for the strict inequality. What do I miss? Is it because it contains an open set?



Any help/hints are much appreciated! Thanks in advance!






real-analysis analysis measure-theory lebesgue-measure







share|cite|improve this question











$endgroup$




Let $f in W^{2,p}(U)$ for $1 le p lt infty$ where $U$ is a compact subset of $mathbb R^2$ and $g in L^{infty}(U)$ with $g(f)=f$ almost everywhere on $U$. I have trouble understanding why the following holds:




Assume that the set ${f=0}$ is non empty and moreover that $g lt 1$ in ${f=0}$. Then it has a strictly
positive lebesgue measure.




I know that ${f=0}$ is a (relatively) closed set but that is not enough for the strict inequality. What do I miss? Is it because it contains an open set?



Any help/hints are much appreciated! Thanks in advance!






real-analysis analysis measure-theory lebesgue-measure




real-analysis analysis measure-theory lebesgue-measure






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 3 '18 at 12:15







kaithkolesidou

















asked Nov 28 '18 at 12:55









kaithkolesidoukaithkolesidou

959511




959511












  • $begingroup$
    Do you not have any additional assumptions? Unless I'm missing something, $f(x) = x_1^2+x_2^2$ is an easy counterexample.
    $endgroup$
    – ktoi
    Nov 28 '18 at 21:26












  • $begingroup$
    @ktoi I just edited my question...
    $endgroup$
    – kaithkolesidou
    Dec 3 '18 at 11:16






  • 1




    $begingroup$
    What is the connection between $f$ and $g$. The example given by ktoi is still valid with $g=0$.
    $endgroup$
    – Kavi Rama Murthy
    Dec 3 '18 at 12:01










  • $begingroup$
    @KaviRamaMurthy I'm sorry I didn't mention it from the beginning. I edited
    $endgroup$
    – kaithkolesidou
    Dec 3 '18 at 12:16










  • $begingroup$
    @kaithkolesidou What do you mean by $g(f)$? If it's functional composition, then the assumption $g<1$ on ${f = 0}$ doesn't really tell you anything and taking $g(x) = x$ my example still remains valid. Also it seems like you need more regularity assumptions on $g$; if ${ f = 0 }$ is a null set then the condition $g < 1$ on ${f = 0}$ gives hardly any restriction on $g.$
    $endgroup$
    – ktoi
    Dec 4 '18 at 19:07


















  • $begingroup$
    Do you not have any additional assumptions? Unless I'm missing something, $f(x) = x_1^2+x_2^2$ is an easy counterexample.
    $endgroup$
    – ktoi
    Nov 28 '18 at 21:26












  • $begingroup$
    @ktoi I just edited my question...
    $endgroup$
    – kaithkolesidou
    Dec 3 '18 at 11:16






  • 1




    $begingroup$
    What is the connection between $f$ and $g$. The example given by ktoi is still valid with $g=0$.
    $endgroup$
    – Kavi Rama Murthy
    Dec 3 '18 at 12:01










  • $begingroup$
    @KaviRamaMurthy I'm sorry I didn't mention it from the beginning. I edited
    $endgroup$
    – kaithkolesidou
    Dec 3 '18 at 12:16










  • $begingroup$
    @kaithkolesidou What do you mean by $g(f)$? If it's functional composition, then the assumption $g<1$ on ${f = 0}$ doesn't really tell you anything and taking $g(x) = x$ my example still remains valid. Also it seems like you need more regularity assumptions on $g$; if ${ f = 0 }$ is a null set then the condition $g < 1$ on ${f = 0}$ gives hardly any restriction on $g.$
    $endgroup$
    – ktoi
    Dec 4 '18 at 19:07
















$begingroup$
Do you not have any additional assumptions? Unless I'm missing something, $f(x) = x_1^2+x_2^2$ is an easy counterexample.
$endgroup$
– ktoi
Nov 28 '18 at 21:26






$begingroup$
Do you not have any additional assumptions? Unless I'm missing something, $f(x) = x_1^2+x_2^2$ is an easy counterexample.
$endgroup$
– ktoi
Nov 28 '18 at 21:26














$begingroup$
@ktoi I just edited my question...
$endgroup$
– kaithkolesidou
Dec 3 '18 at 11:16




$begingroup$
@ktoi I just edited my question...
$endgroup$
– kaithkolesidou
Dec 3 '18 at 11:16




1




1




$begingroup$
What is the connection between $f$ and $g$. The example given by ktoi is still valid with $g=0$.
$endgroup$
– Kavi Rama Murthy
Dec 3 '18 at 12:01




$begingroup$
What is the connection between $f$ and $g$. The example given by ktoi is still valid with $g=0$.
$endgroup$
– Kavi Rama Murthy
Dec 3 '18 at 12:01












$begingroup$
@KaviRamaMurthy I'm sorry I didn't mention it from the beginning. I edited
$endgroup$
– kaithkolesidou
Dec 3 '18 at 12:16




$begingroup$
@KaviRamaMurthy I'm sorry I didn't mention it from the beginning. I edited
$endgroup$
– kaithkolesidou
Dec 3 '18 at 12:16












$begingroup$
@kaithkolesidou What do you mean by $g(f)$? If it's functional composition, then the assumption $g<1$ on ${f = 0}$ doesn't really tell you anything and taking $g(x) = x$ my example still remains valid. Also it seems like you need more regularity assumptions on $g$; if ${ f = 0 }$ is a null set then the condition $g < 1$ on ${f = 0}$ gives hardly any restriction on $g.$
$endgroup$
– ktoi
Dec 4 '18 at 19:07




$begingroup$
@kaithkolesidou What do you mean by $g(f)$? If it's functional composition, then the assumption $g<1$ on ${f = 0}$ doesn't really tell you anything and taking $g(x) = x$ my example still remains valid. Also it seems like you need more regularity assumptions on $g$; if ${ f = 0 }$ is a null set then the condition $g < 1$ on ${f = 0}$ gives hardly any restriction on $g.$
$endgroup$
– ktoi
Dec 4 '18 at 19:07










0






active

oldest

votes











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3017107%2fquestion-on-strictly-positive-measure-of-a-level-set%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























0






active

oldest

votes








0






active

oldest

votes









active

oldest

votes






active

oldest

votes
















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3017107%2fquestion-on-strictly-positive-measure-of-a-level-set%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







-

Popular posts from this blog

Plaza Victoria

Image
Plaza Victoria Plaza Victoria en 2017 Otros nombres Plaza de La Victoria Localización El Almendral, Valparaíso, Chile Vías adyacentes Chacabuco, Molina, Plaza Victoria y Edwards Creación 1841 Metro Bellavista Ubicación 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 Coordenadas: 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 [editar datos en Wikidata] La Plaza Victoria , también llamada Plaza de la Victoria , [ 1 ] ​ es una plaza ubicada en el sector El Almendral del plan de la ciudad de Valparaíso, Chile, de importancia patrimonial, que incluye valiosas estatuas y especies vegetales. [ 2 ] ​ La plaza se creó como tal en la primera mitad del siglo XIX, tomando previamente los nombres de Plaza Nueva y Plaza de Orrego . Si bien desde sus inicios tuvo relevancia histórica, no fue sino a partir de fines de los años 1860 cuando comenzó a cobrar mayor importancia y se convirtió en uno de los principales centros sociales de l...
Read more

Puebla de Zaragoza

Image
Para otros usos de este término, véase Puebla (desambiguación). Heroica Puebla de Zaragoza Localidad De arriba a abajo de izquierda a derecha: Estrella de Puebla, Volcán Popocatépetl, Catedral de Puebla, Monumento a Ignacio Zaragoza, Estadio Cuauhtémoc, Angelópolis y Vista panorámica de la ciudad. Escudo Otros nombres: El relicario de América/ La ciudad de los Ángeles/ La Angelópolis Lema: Muy noble y muy leal ciudad de Puebla de los Ángeles Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en México Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en Puebla Mapa interactivo Coordenadas 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Coordenadas: 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Idioma oficial Español Entidad Localidad  • País   México  • Estado Puebla  • Municipio Puebla Presidente municipal Claudia Rivera Vivanc...
Read more

Musa

Image
Para el género de plantas, véase Musa (planta). Apolo y la Nueve Musas En la mitología griega, las musas (en griego antiguo μοῦσαι « mousai» ) son, según los escritores más antiguos, las divinidades inspiradoras de las Artes: cada una de ellas está relacionada con ramas artísticas y del conocimiento. Son hijas de Zeus y de Mnemósine, compañeras del séquito de Apolo, dios olímpico de la música y patrón de las bellas artes, quien tuvo romances con cada una de ellas, dejando descendientes. Bajaban a la tierra a susurrar ideas e inspirar a aquellos mortales que las invocaran. En la época más arcaica eran las ninfas inspiradoras de las fuentes, en las cuales eran adoradas. Finalmente, alrededor de los siglos VIII-VII a. C. [ 1 ] ​ prevaleció en todo el territorio de la Hélade la adoración de las nueve Musas, que son Calíope, Clío, Erato, Euterpe, Melpómene, Polimnia, Talía, Terpsícore y Urania. El culto a las musas era originalmente de Tracia y Beocia, y fueron de vital impo...
Read more