Borel sets. Need to prove.
-2
I need to prove that set $(1,4)$ is Borel set. Actually, I have no idea how to do it. I was looking for theory. And find something.
Let $I={(1,4), a<b, text{ and } a,b in R}$
Than $(1,4)= cup_{n=1}^infty (1, 4-frac{1}{n}], a<b$.
and it is a proof? How I should prove it?
probability borel-sets
asked Nov 20 at 8:56
Atstovas
697
add a comment |
-2
I need to prove that set $(1,4)$ is Borel set. Actually, I have no idea how to do it. I was looking for theory. And find something.
Let $I={(1,4), a<b, text{ and } a,b in R}$
Than $(1,4)= cup_{n=1}^infty (1, 4-frac{1}{n}], a<b$.
and it is a proof? How I should prove it?
probability borel-sets
asked Nov 20 at 8:56
Atstovas
697
What is your definition of Borel set? It seems that you think of them as elements of $sigma$-algebra generated left-open/right-closed intervals like $(a,b]$. Then what you mention is indeed a proof that $(1,4)$ is a Borel set. That "definition" is okay, but it is better to define them as elements of $sigma$-algebra generated by open sets (as you meet in the answer of Fred). Then your definition becomes a theorem. If it is unclear which definition is practicized then it is not possible to recognize "proofs".
– drhab
Nov 20 at 9:08
add a comment |
-2
-2
-2
I need to prove that set $(1,4)$ is Borel set. Actually, I have no idea how to do it. I was looking for theory. And find something.
Let $I={(1,4), a<b, text{ and } a,b in R}$
Than $(1,4)= cup_{n=1}^infty (1, 4-frac{1}{n}], a<b$.
and it is a proof? How I should prove it?
probability borel-sets
asked Nov 20 at 8:56
Atstovas
697
I need to prove that set $(1,4)$ is Borel set. Actually, I have no idea how to do it. I was looking for theory. And find something.
Let $I={(1,4), a<b, text{ and } a,b in R}$
Than $(1,4)= cup_{n=1}^infty (1, 4-frac{1}{n}], a<b$.
and it is a proof? How I should prove it?
probability borel-sets
probability borel-sets
asked Nov 20 at 8:56
Atstovas
697
asked Nov 20 at 8:56
Atstovas
697
asked Nov 20 at 8:56
Atstovas
697
asked Nov 20 at 8:56
Atstovas
697
asked Nov 20 at 8:56
Atstovas
697
697
What is your definition of Borel set? It seems that you think of them as elements of $sigma$-algebra generated left-open/right-closed intervals like $(a,b]$. Then what you mention is indeed a proof that $(1,4)$ is a Borel set. That "definition" is okay, but it is better to define them as elements of $sigma$-algebra generated by open sets (as you meet in the answer of Fred). Then your definition becomes a theorem. If it is unclear which definition is practicized then it is not possible to recognize "proofs".
– drhab
Nov 20 at 9:08
add a comment |
What is your definition of Borel set? It seems that you think of them as elements of $sigma$-algebra generated left-open/right-closed intervals like $(a,b]$. Then what you mention is indeed a proof that $(1,4)$ is a Borel set. That "definition" is okay, but it is better to define them as elements of $sigma$-algebra generated by open sets (as you meet in the answer of Fred). Then your definition becomes a theorem. If it is unclear which definition is practicized then it is not possible to recognize "proofs".
– drhab
Nov 20 at 9:08
What is your definition of Borel set? It seems that you think of them as elements of $sigma$-algebra generated left-open/right-closed intervals like $(a,b]$. Then what you mention is indeed a proof that $(1,4)$ is a Borel set. That "definition" is okay, but it is better to define them as elements of $sigma$-algebra generated by open sets (as you meet in the answer of Fred). Then your definition becomes a theorem. If it is unclear which definition is practicized then it is not possible to recognize "proofs".
– drhab
Nov 20 at 9:08
What is your definition of Borel set? It seems that you think of them as elements of $sigma$-algebra generated left-open/right-closed intervals like $(a,b]$. Then what you mention is indeed a proof that $(1,4)$ is a Borel set. That "definition" is okay, but it is better to define them as elements of $sigma$-algebra generated by open sets (as you meet in the answer of Fred). Then your definition becomes a theorem. If it is unclear which definition is practicized then it is not possible to recognize "proofs".
– drhab
Nov 20 at 9:08
add a comment |
1 Answer
1
active
oldest
votes
2
The Borel sets in $ mathbb R$ are generated by the open sets in $ mathbb R$ and $(1,4)$ is open !
answered Nov 20 at 8:58
Fred
44.2k1845
Than what about set [1,4]? how to prove it is a Borel set?
– Atstovas
Nov 20 at 9:00
1
The complement of $[1,4]$ is open !
– Fred
Nov 20 at 9:03
and that is it? Enough to show this and nothing more to explain? I didn't think it is that easy...
– Atstovas
Nov 20 at 9:03
If $A$ is a member of a $ sigma-$ algebra, then the complement of $A$ is also a member of the $ sigma-$ algebra
– Fred
Nov 20 at 9:05
Ok, what about $mathbb{N}$ set. How to prove it is a Borel set?
– Atstovas
Nov 20 at 9:05
|
show 1 more 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
});
}
});
draft saved
draft discarded
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%2f3006089%2fborel-sets-need-to-prove%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
2
The Borel sets in $ mathbb R$ are generated by the open sets in $ mathbb R$ and $(1,4)$ is open !
answered Nov 20 at 8:58
Fred
44.2k1845
Than what about set [1,4]? how to prove it is a Borel set?
– Atstovas
Nov 20 at 9:00
1
The complement of $[1,4]$ is open !
– Fred
Nov 20 at 9:03
and that is it? Enough to show this and nothing more to explain? I didn't think it is that easy...
– Atstovas
Nov 20 at 9:03
If $A$ is a member of a $ sigma-$ algebra, then the complement of $A$ is also a member of the $ sigma-$ algebra
– Fred
Nov 20 at 9:05
Ok, what about $mathbb{N}$ set. How to prove it is a Borel set?
– Atstovas
Nov 20 at 9:05
|
show 1 more comment
2
The Borel sets in $ mathbb R$ are generated by the open sets in $ mathbb R$ and $(1,4)$ is open !
answered Nov 20 at 8:58
Fred
44.2k1845
Than what about set [1,4]? how to prove it is a Borel set?
– Atstovas
Nov 20 at 9:00
1
The complement of $[1,4]$ is open !
– Fred
Nov 20 at 9:03
and that is it? Enough to show this and nothing more to explain? I didn't think it is that easy...
– Atstovas
Nov 20 at 9:03
If $A$ is a member of a $ sigma-$ algebra, then the complement of $A$ is also a member of the $ sigma-$ algebra
– Fred
Nov 20 at 9:05
Ok, what about $mathbb{N}$ set. How to prove it is a Borel set?
– Atstovas
Nov 20 at 9:05
|
show 1 more comment
2
2
2
The Borel sets in $ mathbb R$ are generated by the open sets in $ mathbb R$ and $(1,4)$ is open !
answered Nov 20 at 8:58
Fred
44.2k1845
The Borel sets in $ mathbb R$ are generated by the open sets in $ mathbb R$ and $(1,4)$ is open !
answered Nov 20 at 8:58
Fred
44.2k1845
answered Nov 20 at 8:58
Fred
44.2k1845
answered Nov 20 at 8:58
Fred
44.2k1845
answered Nov 20 at 8:58
Fred
44.2k1845
44.2k1845
Than what about set [1,4]? how to prove it is a Borel set?
– Atstovas
Nov 20 at 9:00
1
The complement of $[1,4]$ is open !
– Fred
Nov 20 at 9:03
and that is it? Enough to show this and nothing more to explain? I didn't think it is that easy...
– Atstovas
Nov 20 at 9:03
If $A$ is a member of a $ sigma-$ algebra, then the complement of $A$ is also a member of the $ sigma-$ algebra
– Fred
Nov 20 at 9:05
Ok, what about $mathbb{N}$ set. How to prove it is a Borel set?
– Atstovas
Nov 20 at 9:05
|
show 1 more comment
Than what about set [1,4]? how to prove it is a Borel set?
– Atstovas
Nov 20 at 9:00
1
The complement of $[1,4]$ is open !
– Fred
Nov 20 at 9:03
and that is it? Enough to show this and nothing more to explain? I didn't think it is that easy...
– Atstovas
Nov 20 at 9:03
If $A$ is a member of a $ sigma-$ algebra, then the complement of $A$ is also a member of the $ sigma-$ algebra
– Fred
Nov 20 at 9:05
Ok, what about $mathbb{N}$ set. How to prove it is a Borel set?
– Atstovas
Nov 20 at 9:05
Than what about set [1,4]? how to prove it is a Borel set?
– Atstovas
Nov 20 at 9:00
Than what about set [1,4]? how to prove it is a Borel set?
– Atstovas
Nov 20 at 9:00
1
1
The complement of $[1,4]$ is open !
– Fred
Nov 20 at 9:03
The complement of $[1,4]$ is open !
– Fred
Nov 20 at 9:03
and that is it? Enough to show this and nothing more to explain? I didn't think it is that easy...
– Atstovas
Nov 20 at 9:03
and that is it? Enough to show this and nothing more to explain? I didn't think it is that easy...
– Atstovas
Nov 20 at 9:03
If $A$ is a member of a $ sigma-$ algebra, then the complement of $A$ is also a member of the $ sigma-$ algebra
– Fred
Nov 20 at 9:05
If $A$ is a member of a $ sigma-$ algebra, then the complement of $A$ is also a member of the $ sigma-$ algebra
– Fred
Nov 20 at 9:05
Ok, what about $mathbb{N}$ set. How to prove it is a Borel set?
– Atstovas
Nov 20 at 9:05
Ok, what about $mathbb{N}$ set. How to prove it is a Borel set?
– Atstovas
Nov 20 at 9:05
|
show 1 more comment
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.
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.
draft saved
draft discarded
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%2f3006089%2fborel-sets-need-to-prove%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
What is your definition of Borel set? It seems that you think of them as elements of $sigma$-algebra generated left-open/right-closed intervals like $(a,b]$. Then what you mention is indeed a proof that $(1,4)$ is a Borel set. That "definition" is okay, but it is better to define them as elements of $sigma$-algebra generated by open sets (as you meet in the answer of Fred). Then your definition becomes a theorem. If it is unclear which definition is practicized then it is not possible to recognize "proofs".
– drhab
Nov 20 at 9:08