$mu$, a Borel measure on $mathbb{R}$ satisfies $mu(K)<infty$ for compact K, $mu(x+E)=mu(E)$. Then $mu(E)...
I wish to show that any Borel measure which is finite over compact sets and is invariant under Euclidean isometrics, is in fact Lebesgue's measure multiplied by a positive factor. In other words: $forall E in mathcal{B_{mathbb{R}}} : mu(E) = ccdot lambda(E)$, when $lambda$ is Lebesgue's measure.
I proved that for $c = mu ([0,1])$ and for any open set in $mathbb{R}$. Now defining $mathcal{E} = {E : mu(E) = mu([0,1])cdot lambda(E) }$ I wish to demonstrate that $mathcal{E}$ is a $sigma$ algebra and then $mathcal{B}_{mathbb{R}}subseteqmathcal{E}$.
I don't succeed to show closure under complement, and I would like some help with that.
For non-emptiness and closure under countable unions:
$mathcal{E}$ is not empty , because it contains all open sets.
${E_n}_1^infty$ a disjoint collection in $mathcal{E}$ : $mu(biguplus E_n) = sum_1^inftymu(E_n)=sum_1^infty mu([0,1])cdot lambda(E_n) = mu([0,1])cdot lambda(biguplus E_n)$. Thus $mathcal{E}$ is closed to countable unions (I am using here a lemma which promises it's enough to demonstrate it for disjoint collection).
measure-theory lebesgue-measure
asked Nov 18 at 16:42
dan
470312
add a comment |
I wish to show that any Borel measure which is finite over compact sets and is invariant under Euclidean isometrics, is in fact Lebesgue's measure multiplied by a positive factor. In other words: $forall E in mathcal{B_{mathbb{R}}} : mu(E) = ccdot lambda(E)$, when $lambda$ is Lebesgue's measure.
I proved that for $c = mu ([0,1])$ and for any open set in $mathbb{R}$. Now defining $mathcal{E} = {E : mu(E) = mu([0,1])cdot lambda(E) }$ I wish to demonstrate that $mathcal{E}$ is a $sigma$ algebra and then $mathcal{B}_{mathbb{R}}subseteqmathcal{E}$.
I don't succeed to show closure under complement, and I would like some help with that.
For non-emptiness and closure under countable unions:
$mathcal{E}$ is not empty , because it contains all open sets.
${E_n}_1^infty$ a disjoint collection in $mathcal{E}$ : $mu(biguplus E_n) = sum_1^inftymu(E_n)=sum_1^infty mu([0,1])cdot lambda(E_n) = mu([0,1])cdot lambda(biguplus E_n)$. Thus $mathcal{E}$ is closed to countable unions (I am using here a lemma which promises it's enough to demonstrate it for disjoint collection).
measure-theory lebesgue-measure
asked Nov 18 at 16:42
dan
470312
add a comment |
I wish to show that any Borel measure which is finite over compact sets and is invariant under Euclidean isometrics, is in fact Lebesgue's measure multiplied by a positive factor. In other words: $forall E in mathcal{B_{mathbb{R}}} : mu(E) = ccdot lambda(E)$, when $lambda$ is Lebesgue's measure.
I proved that for $c = mu ([0,1])$ and for any open set in $mathbb{R}$. Now defining $mathcal{E} = {E : mu(E) = mu([0,1])cdot lambda(E) }$ I wish to demonstrate that $mathcal{E}$ is a $sigma$ algebra and then $mathcal{B}_{mathbb{R}}subseteqmathcal{E}$.
I don't succeed to show closure under complement, and I would like some help with that.
For non-emptiness and closure under countable unions:
$mathcal{E}$ is not empty , because it contains all open sets.
${E_n}_1^infty$ a disjoint collection in $mathcal{E}$ : $mu(biguplus E_n) = sum_1^inftymu(E_n)=sum_1^infty mu([0,1])cdot lambda(E_n) = mu([0,1])cdot lambda(biguplus E_n)$. Thus $mathcal{E}$ is closed to countable unions (I am using here a lemma which promises it's enough to demonstrate it for disjoint collection).
measure-theory lebesgue-measure
asked Nov 18 at 16:42
dan
470312
I wish to show that any Borel measure which is finite over compact sets and is invariant under Euclidean isometrics, is in fact Lebesgue's measure multiplied by a positive factor. In other words: $forall E in mathcal{B_{mathbb{R}}} : mu(E) = ccdot lambda(E)$, when $lambda$ is Lebesgue's measure.
I proved that for $c = mu ([0,1])$ and for any open set in $mathbb{R}$. Now defining $mathcal{E} = {E : mu(E) = mu([0,1])cdot lambda(E) }$ I wish to demonstrate that $mathcal{E}$ is a $sigma$ algebra and then $mathcal{B}_{mathbb{R}}subseteqmathcal{E}$.
I don't succeed to show closure under complement, and I would like some help with that.
For non-emptiness and closure under countable unions:
$mathcal{E}$ is not empty , because it contains all open sets.
${E_n}_1^infty$ a disjoint collection in $mathcal{E}$ : $mu(biguplus E_n) = sum_1^inftymu(E_n)=sum_1^infty mu([0,1])cdot lambda(E_n) = mu([0,1])cdot lambda(biguplus E_n)$. Thus $mathcal{E}$ is closed to countable unions (I am using here a lemma which promises it's enough to demonstrate it for disjoint collection).
measure-theory lebesgue-measure
measure-theory lebesgue-measure
asked Nov 18 at 16:42
dan
470312
asked Nov 18 at 16:42
dan
470312
edited Nov 18 at 17:24
asked Nov 18 at 16:42
dan
470312
asked Nov 18 at 16:42
dan
470312
asked Nov 18 at 16:42
dan
470312
470312
add a comment |
add a comment |
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
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3003768%2fmu-a-borel-measure-on-mathbbr-satisfies-muk-infty-for-compact-k%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3003768%2fmu-a-borel-measure-on-mathbbr-satisfies-muk-infty-for-compact-k%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
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
