Mixing
Every finite Borel Measure on a complete separable Metric Space is tight
$begingroup$
I'm trying to understand the proof of the Theorem in the title which I found here
https://www.math.leidenuniv.nl/~vangaans/jancol1.pdf. (Theorem 2.6)
The proof is very good in my opinion however I don't understand why he can choose
$bigcup _{ }^{ infty }{ B({ a }_{ n, }delta )=X } $ in the 4th line.
If I'm not fully mistaken, it doesn't hold in general that every complete metric space can be represented as a union of open balls.
Could someone explain to me why he is allowed to make such a choice?
real-analysis
edited Jan 20 at 20:05
Bernard
122k740116
asked Jan 20 at 19:54
MasterPIMasterPI
21828
$endgroup$
add a comment |
$begingroup$
I'm trying to understand the proof of the Theorem in the title which I found here
https://www.math.leidenuniv.nl/~vangaans/jancol1.pdf. (Theorem 2.6)
The proof is very good in my opinion however I don't understand why he can choose
$bigcup _{ }^{ infty }{ B({ a }_{ n, }delta )=X } $ in the 4th line.
If I'm not fully mistaken, it doesn't hold in general that every complete metric space can be represented as a union of open balls.
Could someone explain to me why he is allowed to make such a choice?
real-analysis
edited Jan 20 at 20:05
Bernard
122k740116
asked Jan 20 at 19:54
MasterPIMasterPI
21828
$endgroup$
$begingroup$
Just as a comment, the title of your question is misleading. Your problem is entirely about topology.
$endgroup$
– alexp9
Jan 20 at 20:18
add a comment |
$begingroup$
I'm trying to understand the proof of the Theorem in the title which I found here
https://www.math.leidenuniv.nl/~vangaans/jancol1.pdf. (Theorem 2.6)
The proof is very good in my opinion however I don't understand why he can choose
$bigcup _{ }^{ infty }{ B({ a }_{ n, }delta )=X } $ in the 4th line.
If I'm not fully mistaken, it doesn't hold in general that every complete metric space can be represented as a union of open balls.
Could someone explain to me why he is allowed to make such a choice?
real-analysis
edited Jan 20 at 20:05
Bernard
122k740116
asked Jan 20 at 19:54
MasterPIMasterPI
21828
$endgroup$
I'm trying to understand the proof of the Theorem in the title which I found here
https://www.math.leidenuniv.nl/~vangaans/jancol1.pdf. (Theorem 2.6)
The proof is very good in my opinion however I don't understand why he can choose
$bigcup _{ }^{ infty }{ B({ a }_{ n, }delta )=X } $ in the 4th line.
If I'm not fully mistaken, it doesn't hold in general that every complete metric space can be represented as a union of open balls.
Could someone explain to me why he is allowed to make such a choice?
real-analysis
real-analysis
edited Jan 20 at 20:05
Bernard
122k740116
asked Jan 20 at 19:54
MasterPIMasterPI
21828
edited Jan 20 at 20:05
Bernard
122k740116
asked Jan 20 at 19:54
MasterPIMasterPI
21828
edited Jan 20 at 20:05
Bernard
122k740116
edited Jan 20 at 20:05
Bernard
122k740116
edited Jan 20 at 20:05
Bernard
122k740116
122k740116
asked Jan 20 at 19:54
MasterPIMasterPI
21828
asked Jan 20 at 19:54
MasterPIMasterPI
21828
asked Jan 20 at 19:54
MasterPIMasterPI
21828
21828
$begingroup$
Just as a comment, the title of your question is misleading. Your problem is entirely about topology.
$endgroup$
– alexp9
Jan 20 at 20:18
add a comment |
$begingroup$
Just as a comment, the title of your question is misleading. Your problem is entirely about topology.
$endgroup$
– alexp9
Jan 20 at 20:18
$begingroup$
Just as a comment, the title of your question is misleading. Your problem is entirely about topology.
$endgroup$
– alexp9
Jan 20 at 20:18
$begingroup$
Just as a comment, the title of your question is misleading. Your problem is entirely about topology.
$endgroup$
– alexp9
Jan 20 at 20:18
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Since it's a metric space, the open balls are well defined. In particular, they exist and $B(a_{n},delta)subset X$ holds. This implies that $bigcup_{i}B(a_{i},delta)subset X.$
Now take $xin X$. Since the sequence of points ${a_{n}}_{ninmathbb{N}}$ is dense in X, $exists iinmathbb{N}$ such that $d(a_{i},x)<frac{delta}{2}$. In particular, $xin B(a_{i},delta)$. Since $xin X$ was arbitrary, we conclude that $Xsubset bigcup_{i} B(a_{i},delta)$.
answered Jan 20 at 20:17
alexp9alexp9
454314
$endgroup$
$begingroup$
Thanks I get it! actually pretty easy
$endgroup$
– MasterPI
Jan 20 at 20:26
add a comment |
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%2f3081070%2fevery-finite-borel-measure-on-a-complete-separable-metric-space-is-tight%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Since it's a metric space, the open balls are well defined. In particular, they exist and $B(a_{n},delta)subset X$ holds. This implies that $bigcup_{i}B(a_{i},delta)subset X.$
Now take $xin X$. Since the sequence of points ${a_{n}}_{ninmathbb{N}}$ is dense in X, $exists iinmathbb{N}$ such that $d(a_{i},x)<frac{delta}{2}$. In particular, $xin B(a_{i},delta)$. Since $xin X$ was arbitrary, we conclude that $Xsubset bigcup_{i} B(a_{i},delta)$.
answered Jan 20 at 20:17
alexp9alexp9
454314
$endgroup$
$begingroup$
Thanks I get it! actually pretty easy
$endgroup$
– MasterPI
Jan 20 at 20:26
add a comment |
$begingroup$
Since it's a metric space, the open balls are well defined. In particular, they exist and $B(a_{n},delta)subset X$ holds. This implies that $bigcup_{i}B(a_{i},delta)subset X.$
Now take $xin X$. Since the sequence of points ${a_{n}}_{ninmathbb{N}}$ is dense in X, $exists iinmathbb{N}$ such that $d(a_{i},x)<frac{delta}{2}$. In particular, $xin B(a_{i},delta)$. Since $xin X$ was arbitrary, we conclude that $Xsubset bigcup_{i} B(a_{i},delta)$.
answered Jan 20 at 20:17
alexp9alexp9
454314
$endgroup$
$begingroup$
Thanks I get it! actually pretty easy
$endgroup$
– MasterPI
Jan 20 at 20:26
add a comment |
$begingroup$
Since it's a metric space, the open balls are well defined. In particular, they exist and $B(a_{n},delta)subset X$ holds. This implies that $bigcup_{i}B(a_{i},delta)subset X.$
Now take $xin X$. Since the sequence of points ${a_{n}}_{ninmathbb{N}}$ is dense in X, $exists iinmathbb{N}$ such that $d(a_{i},x)<frac{delta}{2}$. In particular, $xin B(a_{i},delta)$. Since $xin X$ was arbitrary, we conclude that $Xsubset bigcup_{i} B(a_{i},delta)$.
answered Jan 20 at 20:17
alexp9alexp9
454314
$endgroup$
Since it's a metric space, the open balls are well defined. In particular, they exist and $B(a_{n},delta)subset X$ holds. This implies that $bigcup_{i}B(a_{i},delta)subset X.$
Now take $xin X$. Since the sequence of points ${a_{n}}_{ninmathbb{N}}$ is dense in X, $exists iinmathbb{N}$ such that $d(a_{i},x)<frac{delta}{2}$. In particular, $xin B(a_{i},delta)$. Since $xin X$ was arbitrary, we conclude that $Xsubset bigcup_{i} B(a_{i},delta)$.
answered Jan 20 at 20:17
alexp9alexp9
454314
answered Jan 20 at 20:17
alexp9alexp9
454314
answered Jan 20 at 20:17
alexp9alexp9
454314
answered Jan 20 at 20:17
alexp9alexp9
454314
454314
$begingroup$
Thanks I get it! actually pretty easy
$endgroup$
– MasterPI
Jan 20 at 20:26
add a comment |
$begingroup$
Thanks I get it! actually pretty easy
$endgroup$
– MasterPI
Jan 20 at 20:26
$begingroup$
Thanks I get it! actually pretty easy
$endgroup$
– MasterPI
Jan 20 at 20:26
$begingroup$
Thanks I get it! actually pretty easy
$endgroup$
– MasterPI
Jan 20 at 20:26
add a comment |
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.
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%2f3081070%2fevery-finite-borel-measure-on-a-complete-separable-metric-space-is-tight%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
$begingroup$
Just as a comment, the title of your question is misleading. Your problem is entirely about topology.
$endgroup$
– alexp9
Jan 20 at 20:18