$nu(A)=int mathcal{X}_{A}f(x)dmu$ measure associated functional $l_{nu}(g)=int g(x)f(x)dmu.$
Let $mu$ be a positive Baire measure, $fin L^1(X,dmu)$ with $X$ compact hausdorff and $fgeq 0$.
For any Baire set $A$, define $nu(A)=int mathcal{X}_{A}f(x)dmu$.
Show that $nu$ is the one measure associated to the functional on $C(X)$,
$l_{nu}(g)=int g(x)f(x)dmu.$
Hello! In general, what are the steps to demonstrate that a measure comes from a functional one?
functional-analysis measure-theory
asked Nov 19 '18 at 6:21
eraldcoil
358111
add a comment |
Let $mu$ be a positive Baire measure, $fin L^1(X,dmu)$ with $X$ compact hausdorff and $fgeq 0$.
For any Baire set $A$, define $nu(A)=int mathcal{X}_{A}f(x)dmu$.
Show that $nu$ is the one measure associated to the functional on $C(X)$,
$l_{nu}(g)=int g(x)f(x)dmu.$
Hello! In general, what are the steps to demonstrate that a measure comes from a functional one?
functional-analysis measure-theory
asked Nov 19 '18 at 6:21
eraldcoil
358111
1
There is probably a theorem that gives you uniqueness of measures if the two measures coincide in a set of functions. And that would do it.
– Will M.
Nov 19 '18 at 6:26
1
You are supposed to show that $l_{nu} g=int g dnu$ for all $g in C(X)$. In other words you have to show that $int g(x)f(x)dmu(x)=int g(x)dnu(x)$.
– Kavi Rama Murthy
Nov 19 '18 at 6:27
add a comment |
Let $mu$ be a positive Baire measure, $fin L^1(X,dmu)$ with $X$ compact hausdorff and $fgeq 0$.
For any Baire set $A$, define $nu(A)=int mathcal{X}_{A}f(x)dmu$.
Show that $nu$ is the one measure associated to the functional on $C(X)$,
$l_{nu}(g)=int g(x)f(x)dmu.$
Hello! In general, what are the steps to demonstrate that a measure comes from a functional one?
functional-analysis measure-theory
asked Nov 19 '18 at 6:21
eraldcoil
358111
Let $mu$ be a positive Baire measure, $fin L^1(X,dmu)$ with $X$ compact hausdorff and $fgeq 0$.
For any Baire set $A$, define $nu(A)=int mathcal{X}_{A}f(x)dmu$.
Show that $nu$ is the one measure associated to the functional on $C(X)$,
$l_{nu}(g)=int g(x)f(x)dmu.$
Hello! In general, what are the steps to demonstrate that a measure comes from a functional one?
functional-analysis measure-theory
functional-analysis measure-theory
asked Nov 19 '18 at 6:21
eraldcoil
358111
asked Nov 19 '18 at 6:21
eraldcoil
358111
asked Nov 19 '18 at 6:21
eraldcoil
358111
asked Nov 19 '18 at 6:21
eraldcoil
358111
asked Nov 19 '18 at 6:21
eraldcoil
358111
358111
1
There is probably a theorem that gives you uniqueness of measures if the two measures coincide in a set of functions. And that would do it.
– Will M.
Nov 19 '18 at 6:26
1
You are supposed to show that $l_{nu} g=int g dnu$ for all $g in C(X)$. In other words you have to show that $int g(x)f(x)dmu(x)=int g(x)dnu(x)$.
– Kavi Rama Murthy
Nov 19 '18 at 6:27
add a comment |
1
There is probably a theorem that gives you uniqueness of measures if the two measures coincide in a set of functions. And that would do it.
– Will M.
Nov 19 '18 at 6:26
1
You are supposed to show that $l_{nu} g=int g dnu$ for all $g in C(X)$. In other words you have to show that $int g(x)f(x)dmu(x)=int g(x)dnu(x)$.
– Kavi Rama Murthy
Nov 19 '18 at 6:27
1
1
There is probably a theorem that gives you uniqueness of measures if the two measures coincide in a set of functions. And that would do it.
– Will M.
Nov 19 '18 at 6:26
There is probably a theorem that gives you uniqueness of measures if the two measures coincide in a set of functions. And that would do it.
– Will M.
Nov 19 '18 at 6:26
1
1
You are supposed to show that $l_{nu} g=int g dnu$ for all $g in C(X)$. In other words you have to show that $int g(x)f(x)dmu(x)=int g(x)dnu(x)$.
– Kavi Rama Murthy
Nov 19 '18 at 6:27
You are supposed to show that $l_{nu} g=int g dnu$ for all $g in C(X)$. In other words you have to show that $int g(x)f(x)dmu(x)=int g(x)dnu(x)$.
– Kavi Rama Murthy
Nov 19 '18 at 6:27
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%2f3004587%2fnua-int-mathcalx-afxd-mu-measure-associated-functional-l-nug%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%2f3004587%2fnua-int-mathcalx-afxd-mu-measure-associated-functional-l-nug%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
1
There is probably a theorem that gives you uniqueness of measures if the two measures coincide in a set of functions. And that would do it.
– Will M.
Nov 19 '18 at 6:26
1
You are supposed to show that $l_{nu} g=int g dnu$ for all $g in C(X)$. In other words you have to show that $int g(x)f(x)dmu(x)=int g(x)dnu(x)$.
– Kavi Rama Murthy
Nov 19 '18 at 6:27