Clarification regarding the definition of π- and d-systems
$begingroup$
This is taken form A. Karr probability book.
a) A d-system is a family of subsets containing Ω and closed under proper difference (if A, B ∈ D and A ⊆ B, then BA ∈ D) and countable increasing union.
b) A π-system is a family of subsets closed under finite intersection.
A σ-algebra is a d-system and a π-system. In addition, a class that is both a π-system and a d-system is a σ-algebra.
It seems to me that from a) follows b), because "containing Ω and closed under proper difference" imply "closed under complement", and "closed under countable increasing union" imply "closed under finite (and countable) union". Am I missing something? Because, if I'm right, I don’t' understand the need of defining both σ-algebra and d-system.
probability
asked Dec 10 '18 at 5:31
Markus SteinerMarkus Steiner
1036
$endgroup$
add a comment |
$begingroup$
This is taken form A. Karr probability book.
a) A d-system is a family of subsets containing Ω and closed under proper difference (if A, B ∈ D and A ⊆ B, then BA ∈ D) and countable increasing union.
b) A π-system is a family of subsets closed under finite intersection.
A σ-algebra is a d-system and a π-system. In addition, a class that is both a π-system and a d-system is a σ-algebra.
It seems to me that from a) follows b), because "containing Ω and closed under proper difference" imply "closed under complement", and "closed under countable increasing union" imply "closed under finite (and countable) union". Am I missing something? Because, if I'm right, I don’t' understand the need of defining both σ-algebra and d-system.
probability
asked Dec 10 '18 at 5:31
Markus SteinerMarkus Steiner
1036
$endgroup$
$begingroup$
Usually, we are interested in $sigma$-algebras (both in measure theory and probablity theory). You are indeed correct, but I think your reasoning is lacking in detail. See (here)[math.stackexchange.com/questions/991804/… for a complete answer. The motivation (as far as I know) is the other way around. We are only interested in $pi$-systems and $d$-systems because we are interested in $sigma$-algebras.
$endgroup$
– Quoka
Dec 10 '18 at 5:52
add a comment |
$begingroup$
This is taken form A. Karr probability book.
a) A d-system is a family of subsets containing Ω and closed under proper difference (if A, B ∈ D and A ⊆ B, then BA ∈ D) and countable increasing union.
b) A π-system is a family of subsets closed under finite intersection.
A σ-algebra is a d-system and a π-system. In addition, a class that is both a π-system and a d-system is a σ-algebra.
It seems to me that from a) follows b), because "containing Ω and closed under proper difference" imply "closed under complement", and "closed under countable increasing union" imply "closed under finite (and countable) union". Am I missing something? Because, if I'm right, I don’t' understand the need of defining both σ-algebra and d-system.
probability
asked Dec 10 '18 at 5:31
Markus SteinerMarkus Steiner
1036
$endgroup$
This is taken form A. Karr probability book.
a) A d-system is a family of subsets containing Ω and closed under proper difference (if A, B ∈ D and A ⊆ B, then BA ∈ D) and countable increasing union.
b) A π-system is a family of subsets closed under finite intersection.
A σ-algebra is a d-system and a π-system. In addition, a class that is both a π-system and a d-system is a σ-algebra.
It seems to me that from a) follows b), because "containing Ω and closed under proper difference" imply "closed under complement", and "closed under countable increasing union" imply "closed under finite (and countable) union". Am I missing something? Because, if I'm right, I don’t' understand the need of defining both σ-algebra and d-system.
probability
probability
asked Dec 10 '18 at 5:31
Markus SteinerMarkus Steiner
1036
asked Dec 10 '18 at 5:31
Markus SteinerMarkus Steiner
1036
asked Dec 10 '18 at 5:31
Markus SteinerMarkus Steiner
1036
asked Dec 10 '18 at 5:31
Markus SteinerMarkus Steiner
1036
asked Dec 10 '18 at 5:31
Markus SteinerMarkus Steiner
1036
1036
$begingroup$
Usually, we are interested in $sigma$-algebras (both in measure theory and probablity theory). You are indeed correct, but I think your reasoning is lacking in detail. See (here)[math.stackexchange.com/questions/991804/… for a complete answer. The motivation (as far as I know) is the other way around. We are only interested in $pi$-systems and $d$-systems because we are interested in $sigma$-algebras.
$endgroup$
– Quoka
Dec 10 '18 at 5:52
add a comment |
$begingroup$
Usually, we are interested in $sigma$-algebras (both in measure theory and probablity theory). You are indeed correct, but I think your reasoning is lacking in detail. See (here)[math.stackexchange.com/questions/991804/… for a complete answer. The motivation (as far as I know) is the other way around. We are only interested in $pi$-systems and $d$-systems because we are interested in $sigma$-algebras.
$endgroup$
– Quoka
Dec 10 '18 at 5:52
$begingroup$
Usually, we are interested in $sigma$-algebras (both in measure theory and probablity theory). You are indeed correct, but I think your reasoning is lacking in detail. See (here)[math.stackexchange.com/questions/991804/… for a complete answer. The motivation (as far as I know) is the other way around. We are only interested in $pi$-systems and $d$-systems because we are interested in $sigma$-algebras.
$endgroup$
– Quoka
Dec 10 '18 at 5:52
$begingroup$
Usually, we are interested in $sigma$-algebras (both in measure theory and probablity theory). You are indeed correct, but I think your reasoning is lacking in detail. See (here)[math.stackexchange.com/questions/991804/… for a complete answer. The motivation (as far as I know) is the other way around. We are only interested in $pi$-systems and $d$-systems because we are interested in $sigma$-algebras.
$endgroup$
– Quoka
Dec 10 '18 at 5:52
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Upon reflecting more on this I think I found the mistake in my reasoning. It's actually false that "closed under countable increasing union" imply "closed under finite (and countable) union". In fact you need closed under finite union to write an arbitrary sequence of events as an increasing sequence. So that is the key difference between d-system and σ-algebra. And this is what I was missing on my first reading of that passage of Karr's book.
answered Dec 10 '18 at 7:57
Markus SteinerMarkus Steiner
1036
$endgroup$
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%2f3033493%2fclarification-regarding-the-definition-of-%25cf%2580-and-d-systems%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
$begingroup$
Upon reflecting more on this I think I found the mistake in my reasoning. It's actually false that "closed under countable increasing union" imply "closed under finite (and countable) union". In fact you need closed under finite union to write an arbitrary sequence of events as an increasing sequence. So that is the key difference between d-system and σ-algebra. And this is what I was missing on my first reading of that passage of Karr's book.
answered Dec 10 '18 at 7:57
Markus SteinerMarkus Steiner
1036
$endgroup$
add a comment |
$begingroup$
Upon reflecting more on this I think I found the mistake in my reasoning. It's actually false that "closed under countable increasing union" imply "closed under finite (and countable) union". In fact you need closed under finite union to write an arbitrary sequence of events as an increasing sequence. So that is the key difference between d-system and σ-algebra. And this is what I was missing on my first reading of that passage of Karr's book.
answered Dec 10 '18 at 7:57
Markus SteinerMarkus Steiner
1036
$endgroup$
add a comment |
$begingroup$
Upon reflecting more on this I think I found the mistake in my reasoning. It's actually false that "closed under countable increasing union" imply "closed under finite (and countable) union". In fact you need closed under finite union to write an arbitrary sequence of events as an increasing sequence. So that is the key difference between d-system and σ-algebra. And this is what I was missing on my first reading of that passage of Karr's book.
answered Dec 10 '18 at 7:57
Markus SteinerMarkus Steiner
1036
$endgroup$
Upon reflecting more on this I think I found the mistake in my reasoning. It's actually false that "closed under countable increasing union" imply "closed under finite (and countable) union". In fact you need closed under finite union to write an arbitrary sequence of events as an increasing sequence. So that is the key difference between d-system and σ-algebra. And this is what I was missing on my first reading of that passage of Karr's book.
answered Dec 10 '18 at 7:57
Markus SteinerMarkus Steiner
1036
answered Dec 10 '18 at 7:57
Markus SteinerMarkus Steiner
1036
answered Dec 10 '18 at 7:57
Markus SteinerMarkus Steiner
1036
answered Dec 10 '18 at 7:57
Markus SteinerMarkus Steiner
1036
1036
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%2f3033493%2fclarification-regarding-the-definition-of-%25cf%2580-and-d-systems%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
$begingroup$
Usually, we are interested in $sigma$-algebras (both in measure theory and probablity theory). You are indeed correct, but I think your reasoning is lacking in detail. See (here)[math.stackexchange.com/questions/991804/… for a complete answer. The motivation (as far as I know) is the other way around. We are only interested in $pi$-systems and $d$-systems because we are interested in $sigma$-algebras.
$endgroup$
– Quoka
Dec 10 '18 at 5:52