Example of a set which is not in the product $sigma$-algebra
$begingroup$
Let $L_d$ be the $sigma$-algebra of Lebesgue measurable subsets of $mathbb{R}^d$.
By using Vitali's set $E subseteq [0,1]$, I am looking for an example of $A in L_2$ which is not in the product $sigma$-algebra $L_1 times L_1$.
I am also having trouble proving that $L_1 times L_1 subseteq L_2$. I can see that we can use $mathcal{B}(mathbb{R^2})=mathcal{B}(mathbb{R}) times mathcal{B}(mathbb{R})$ and that the Lebesgue measure $lambda_2$ on $(mathbb{R^2},mathcal{B}(mathbb{R^2}))$ is identical to the product measure $lambda_1 times lambda_1.$ Although I'm stuck afterwards.
measure-theory lebesgue-measure
asked Dec 5 '18 at 20:57
MilTomMilTom
1288
$endgroup$
add a comment |
$begingroup$
Let $L_d$ be the $sigma$-algebra of Lebesgue measurable subsets of $mathbb{R}^d$.
By using Vitali's set $E subseteq [0,1]$, I am looking for an example of $A in L_2$ which is not in the product $sigma$-algebra $L_1 times L_1$.
I am also having trouble proving that $L_1 times L_1 subseteq L_2$. I can see that we can use $mathcal{B}(mathbb{R^2})=mathcal{B}(mathbb{R}) times mathcal{B}(mathbb{R})$ and that the Lebesgue measure $lambda_2$ on $(mathbb{R^2},mathcal{B}(mathbb{R^2}))$ is identical to the product measure $lambda_1 times lambda_1.$ Although I'm stuck afterwards.
measure-theory lebesgue-measure
asked Dec 5 '18 at 20:57
MilTomMilTom
1288
$endgroup$
add a comment |
$begingroup$
Let $L_d$ be the $sigma$-algebra of Lebesgue measurable subsets of $mathbb{R}^d$.
By using Vitali's set $E subseteq [0,1]$, I am looking for an example of $A in L_2$ which is not in the product $sigma$-algebra $L_1 times L_1$.
I am also having trouble proving that $L_1 times L_1 subseteq L_2$. I can see that we can use $mathcal{B}(mathbb{R^2})=mathcal{B}(mathbb{R}) times mathcal{B}(mathbb{R})$ and that the Lebesgue measure $lambda_2$ on $(mathbb{R^2},mathcal{B}(mathbb{R^2}))$ is identical to the product measure $lambda_1 times lambda_1.$ Although I'm stuck afterwards.
measure-theory lebesgue-measure
asked Dec 5 '18 at 20:57
MilTomMilTom
1288
$endgroup$
Let $L_d$ be the $sigma$-algebra of Lebesgue measurable subsets of $mathbb{R}^d$.
By using Vitali's set $E subseteq [0,1]$, I am looking for an example of $A in L_2$ which is not in the product $sigma$-algebra $L_1 times L_1$.
I am also having trouble proving that $L_1 times L_1 subseteq L_2$. I can see that we can use $mathcal{B}(mathbb{R^2})=mathcal{B}(mathbb{R}) times mathcal{B}(mathbb{R})$ and that the Lebesgue measure $lambda_2$ on $(mathbb{R^2},mathcal{B}(mathbb{R^2}))$ is identical to the product measure $lambda_1 times lambda_1.$ Although I'm stuck afterwards.
measure-theory lebesgue-measure
measure-theory lebesgue-measure
asked Dec 5 '18 at 20:57
MilTomMilTom
1288
asked Dec 5 '18 at 20:57
MilTomMilTom
1288
asked Dec 5 '18 at 20:57
MilTomMilTom
1288
asked Dec 5 '18 at 20:57
MilTomMilTom
1288
asked Dec 5 '18 at 20:57
MilTomMilTom
1288
1288
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
$L_1otimes L_1$ is generated by $mathcal{C}={Atimes B:A,Bin L_1}$. Since $mathcal{C}subset L_2$, $L_1otimes L_1subseteq L_2$.
For a set $Nin L_1$ s.t. $Nne emptyset$ and $lambda_1(N)=0$, the set $Etimes Nin L_2$ ($because Etimes Nsubset [0,1]times N$ and $lambda_2([0,1]times N )=0$) but not in $L_1otimes L_1$ ($because$ for any $L_1otimes L_1$-measurable set $A$, the sections $A^y={xin mathbb{R}:(x,y)in A}$ belong to $L_1$).
answered Dec 6 '18 at 6:37
d.k.o.d.k.o.
9,105628
$endgroup$
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%2f3027642%2fexample-of-a-set-which-is-not-in-the-product-sigma-algebra%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
$begingroup$
$L_1otimes L_1$ is generated by $mathcal{C}={Atimes B:A,Bin L_1}$. Since $mathcal{C}subset L_2$, $L_1otimes L_1subseteq L_2$.
For a set $Nin L_1$ s.t. $Nne emptyset$ and $lambda_1(N)=0$, the set $Etimes Nin L_2$ ($because Etimes Nsubset [0,1]times N$ and $lambda_2([0,1]times N )=0$) but not in $L_1otimes L_1$ ($because$ for any $L_1otimes L_1$-measurable set $A$, the sections $A^y={xin mathbb{R}:(x,y)in A}$ belong to $L_1$).
answered Dec 6 '18 at 6:37
d.k.o.d.k.o.
9,105628
$endgroup$
add a comment |
$begingroup$
$L_1otimes L_1$ is generated by $mathcal{C}={Atimes B:A,Bin L_1}$. Since $mathcal{C}subset L_2$, $L_1otimes L_1subseteq L_2$.
For a set $Nin L_1$ s.t. $Nne emptyset$ and $lambda_1(N)=0$, the set $Etimes Nin L_2$ ($because Etimes Nsubset [0,1]times N$ and $lambda_2([0,1]times N )=0$) but not in $L_1otimes L_1$ ($because$ for any $L_1otimes L_1$-measurable set $A$, the sections $A^y={xin mathbb{R}:(x,y)in A}$ belong to $L_1$).
answered Dec 6 '18 at 6:37
d.k.o.d.k.o.
9,105628
$endgroup$
add a comment |
$begingroup$
$L_1otimes L_1$ is generated by $mathcal{C}={Atimes B:A,Bin L_1}$. Since $mathcal{C}subset L_2$, $L_1otimes L_1subseteq L_2$.
For a set $Nin L_1$ s.t. $Nne emptyset$ and $lambda_1(N)=0$, the set $Etimes Nin L_2$ ($because Etimes Nsubset [0,1]times N$ and $lambda_2([0,1]times N )=0$) but not in $L_1otimes L_1$ ($because$ for any $L_1otimes L_1$-measurable set $A$, the sections $A^y={xin mathbb{R}:(x,y)in A}$ belong to $L_1$).
answered Dec 6 '18 at 6:37
d.k.o.d.k.o.
9,105628
$endgroup$
$L_1otimes L_1$ is generated by $mathcal{C}={Atimes B:A,Bin L_1}$. Since $mathcal{C}subset L_2$, $L_1otimes L_1subseteq L_2$.
For a set $Nin L_1$ s.t. $Nne emptyset$ and $lambda_1(N)=0$, the set $Etimes Nin L_2$ ($because Etimes Nsubset [0,1]times N$ and $lambda_2([0,1]times N )=0$) but not in $L_1otimes L_1$ ($because$ for any $L_1otimes L_1$-measurable set $A$, the sections $A^y={xin mathbb{R}:(x,y)in A}$ belong to $L_1$).
answered Dec 6 '18 at 6:37
d.k.o.d.k.o.
9,105628
answered Dec 6 '18 at 6:37
d.k.o.d.k.o.
9,105628
answered Dec 6 '18 at 6:37
d.k.o.d.k.o.
9,105628
answered Dec 6 '18 at 6:37
d.k.o.d.k.o.
9,105628
9,105628
add a comment |
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%2f3027642%2fexample-of-a-set-which-is-not-in-the-product-sigma-algebra%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
