Definition of map preserve the measure : why is it $mu(T^-1(A))=mu(A)$ and not $mu(T(A))=mu(A)$.Problem with the definition of Lebesgue Measure and borel sets!When is the composition operator assigned to a measure-preserving map unitary?Definition of doubling measureDoes changing the values of a $mathbb P$-a.s. defined map at countably many points preserve the map?Do differentiable functions preserve measure zero sets? Measurable sets?Why do we usually construct the Lebesgue Measure on finite outer measure sets before arbitrary measurable sets?Why do we need to declare a probability measure for the definition of stochastic processes?Why the probability distribution of a uniform random variable is the Lebesgue measure?Making sense of measure-theoretic definition of random variableIs the integral of a continuous map always a Radon measure?
Does this property of comaximal ideals always holds?
Why do Australian milk farmers need to protest supermarkets' milk price?
Do I need life insurance if I can cover my own funeral costs?
PlotLabels with equations not expressions
Science-fiction short story where space navy wanted hospital ships and settlers had guns mounted everywhere
Will a pinhole camera work with instant film?
When do we add an hyphen (-) to a complex adjective word?
Is it possible / allowed to upcast ritual spells?
Does splitting a potentially monolithic application into several smaller ones help prevent bugs?
Why are the outputs of printf and std::cout different
My adviser wants to be the first author
How to deal with a cynical class?
Possible Leak In Concrete
Is it possible that AIC = BIC?
Be in awe of my brilliance!
What are some nice/clever ways to introduce the tonic's dominant seventh chord?
An Accountant Seeks the Help of a Mathematician
2D counterpart of std::array in C++17
Is having access to past exams cheating and, if yes, could it be proven just by a good grade?
How could a scammer know the apps on my phone / iTunes account?
Why using two cd commands in bash script does not execute the second command
Identifying the interval from A♭ to D♯
Have researchers managed to "reverse time"? If so, what does that mean for physics?
Why do passenger jet manufacturers design their planes with stall prevention systems?
Definition of map preserve the measure : why is it $mu(T^-1(A))=mu(A)$ and not $mu(T(A))=mu(A)$.
Problem with the definition of Lebesgue Measure and borel sets!When is the composition operator assigned to a measure-preserving map unitary?Definition of doubling measureDoes changing the values of a $mathbb P$-a.s. defined map at countably many points preserve the map?Do differentiable functions preserve measure zero sets? Measurable sets?Why do we usually construct the Lebesgue Measure on finite outer measure sets before arbitrary measurable sets?Why do we need to declare a probability measure for the definition of stochastic processes?Why the probability distribution of a uniform random variable is the Lebesgue measure?Making sense of measure-theoretic definition of random variableIs the integral of a continuous map always a Radon measure?
$begingroup$
Let $(X, mathcal B,mathbb P)$ a probability space. In wikipedia, they say that a map $T:Xto X$ preserve the measure if $$mathbb P(T^-1(A))=mu(A),$$
for all $Ain mathcal B$.
Why a definition as $mathbb P(T(A))=mathbb P(A)$ would not work ? It looks more natural, no ?
measure-theory
asked Mar 11 at 10:09
PierrePierre
618
New contributor
Pierre is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
Let $(X, mathcal B,mathbb P)$ a probability space. In wikipedia, they say that a map $T:Xto X$ preserve the measure if $$mathbb P(T^-1(A))=mu(A),$$
for all $Ain mathcal B$.
Why a definition as $mathbb P(T(A))=mathbb P(A)$ would not work ? It looks more natural, no ?
measure-theory
asked Mar 11 at 10:09
PierrePierre
618
New contributor
Pierre is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
Let $(X, mathcal B,mathbb P)$ a probability space. In wikipedia, they say that a map $T:Xto X$ preserve the measure if $$mathbb P(T^-1(A))=mu(A),$$
for all $Ain mathcal B$.
Why a definition as $mathbb P(T(A))=mathbb P(A)$ would not work ? It looks more natural, no ?
measure-theory
asked Mar 11 at 10:09
PierrePierre
618
New contributor
Pierre is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Let $(X, mathcal B,mathbb P)$ a probability space. In wikipedia, they say that a map $T:Xto X$ preserve the measure if $$mathbb P(T^-1(A))=mu(A),$$
for all $Ain mathcal B$.
Why a definition as $mathbb P(T(A))=mathbb P(A)$ would not work ? It looks more natural, no ?
measure-theory
measure-theory
asked Mar 11 at 10:09
PierrePierre
618
New contributor
Pierre is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Mar 11 at 10:09
PierrePierre
618
New contributor
Pierre is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Mar 11 at 10:09
PierrePierre
618
New contributor
Pierre is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Mar 11 at 10:09
PierrePierre
618
asked Mar 11 at 10:09
PierrePierre
618
618
New contributor
Pierre is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Pierre is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Pierre is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The image of a measurabe set under a measurable transformation may not be measurable but the inverse image is always measurable (by definition of measurability).
If $T$ is bijective and $T^-1$ is also measurable thee the two conditions are identical.
answered Mar 11 at 10:12
Kavi Rama MurthyKavi Rama Murthy
67.2k53067
$endgroup$
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
);
);
Pierre is a new contributor. Be nice, and check out our Code of Conduct.
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%2f3143512%2fdefinition-of-map-preserve-the-measure-why-is-it-mut-1a-mua-and-n%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
$begingroup$
The image of a measurabe set under a measurable transformation may not be measurable but the inverse image is always measurable (by definition of measurability).
If $T$ is bijective and $T^-1$ is also measurable thee the two conditions are identical.
answered Mar 11 at 10:12
Kavi Rama MurthyKavi Rama Murthy
67.2k53067
$endgroup$
add a comment |
$begingroup$
The image of a measurabe set under a measurable transformation may not be measurable but the inverse image is always measurable (by definition of measurability).
If $T$ is bijective and $T^-1$ is also measurable thee the two conditions are identical.
answered Mar 11 at 10:12
Kavi Rama MurthyKavi Rama Murthy
67.2k53067
$endgroup$
add a comment |
$begingroup$
The image of a measurabe set under a measurable transformation may not be measurable but the inverse image is always measurable (by definition of measurability).
If $T$ is bijective and $T^-1$ is also measurable thee the two conditions are identical.
answered Mar 11 at 10:12
Kavi Rama MurthyKavi Rama Murthy
67.2k53067
$endgroup$
The image of a measurabe set under a measurable transformation may not be measurable but the inverse image is always measurable (by definition of measurability).
If $T$ is bijective and $T^-1$ is also measurable thee the two conditions are identical.
answered Mar 11 at 10:12
Kavi Rama MurthyKavi Rama Murthy
67.2k53067
answered Mar 11 at 10:12
Kavi Rama MurthyKavi Rama Murthy
67.2k53067
answered Mar 11 at 10:12
Kavi Rama MurthyKavi Rama Murthy
67.2k53067
answered Mar 11 at 10:12
Kavi Rama MurthyKavi Rama Murthy
67.2k53067
67.2k53067
add a comment |
add a comment |
Pierre is a new contributor. Be nice, and check out our Code of Conduct.
Pierre is a new contributor. Be nice, and check out our Code of Conduct.
Pierre is a new contributor. Be nice, and check out our Code of Conduct.
Pierre is a new contributor. Be nice, and check out our Code of Conduct.
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%2f3143512%2fdefinition-of-map-preserve-the-measure-why-is-it-mut-1a-mua-and-n%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
