Is it true $P(cap_{n=1}^infty cup_{m=n}^infty A_m)=lim_{n to infty}P(cup_{m=n}^infty A_m)$?
$begingroup$
I just need some confirmation on the truth of this since I didn't see it in a textbook so far. I think it is true by continuity of probability. Am I right?
probability measure-theory
asked Dec 6 '18 at 3:16
Daniel LiDaniel Li
664413
$endgroup$
add a comment |
$begingroup$
I just need some confirmation on the truth of this since I didn't see it in a textbook so far. I think it is true by continuity of probability. Am I right?
probability measure-theory
asked Dec 6 '18 at 3:16
Daniel LiDaniel Li
664413
$endgroup$
add a comment |
$begingroup$
I just need some confirmation on the truth of this since I didn't see it in a textbook so far. I think it is true by continuity of probability. Am I right?
probability measure-theory
asked Dec 6 '18 at 3:16
Daniel LiDaniel Li
664413
$endgroup$
I just need some confirmation on the truth of this since I didn't see it in a textbook so far. I think it is true by continuity of probability. Am I right?
probability measure-theory
probability measure-theory
asked Dec 6 '18 at 3:16
Daniel LiDaniel Li
664413
asked Dec 6 '18 at 3:16
Daniel LiDaniel Li
664413
asked Dec 6 '18 at 3:16
Daniel LiDaniel Li
664413
asked Dec 6 '18 at 3:16
Daniel LiDaniel Li
664413
asked Dec 6 '18 at 3:16
Daniel LiDaniel Li
664413
664413
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
For each $n in mathbb N$, let $B_n=bigcup_{m=n}^infty A_m $. Clearly for each $n in mathbb N$ we have $B_n supseteq B_{n+1}$, and so ${B_n}_{n=1}^infty$ is a decreasing sequence of sets. Moreover, $P(B_1)<infty$ as a probability measure is finite. Therefore, $$Pleft(bigcap_{n=1}^infty B_nright)=lim_{nto infty} P(B_n)$$ by the continuity from above property of a measure which is $(3')$ here.
answered Dec 6 '18 at 3:38
Matt A PeltoMatt A Pelto
2,537620
$endgroup$
add a comment |
$begingroup$
Let $B_n:=bigcup_{mge n}A_m$. Then $B_{n+1}subseteq B_n$ and by continuity from above,
$$
mathsf{P}left(bigcap_{nge 1} B_nright)=lim_{ntoinfty} mathsf{P}(B_n).
$$
answered Dec 6 '18 at 3:32
d.k.o.d.k.o.
9,145628
$endgroup$
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%2f3027990%2fis-it-true-p-cap-n-1-infty-cup-m-n-infty-a-m-lim-n-to-inftyp-cup%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
For each $n in mathbb N$, let $B_n=bigcup_{m=n}^infty A_m $. Clearly for each $n in mathbb N$ we have $B_n supseteq B_{n+1}$, and so ${B_n}_{n=1}^infty$ is a decreasing sequence of sets. Moreover, $P(B_1)<infty$ as a probability measure is finite. Therefore, $$Pleft(bigcap_{n=1}^infty B_nright)=lim_{nto infty} P(B_n)$$ by the continuity from above property of a measure which is $(3')$ here.
answered Dec 6 '18 at 3:38
Matt A PeltoMatt A Pelto
2,537620
$endgroup$
add a comment |
$begingroup$
For each $n in mathbb N$, let $B_n=bigcup_{m=n}^infty A_m $. Clearly for each $n in mathbb N$ we have $B_n supseteq B_{n+1}$, and so ${B_n}_{n=1}^infty$ is a decreasing sequence of sets. Moreover, $P(B_1)<infty$ as a probability measure is finite. Therefore, $$Pleft(bigcap_{n=1}^infty B_nright)=lim_{nto infty} P(B_n)$$ by the continuity from above property of a measure which is $(3')$ here.
answered Dec 6 '18 at 3:38
Matt A PeltoMatt A Pelto
2,537620
$endgroup$
add a comment |
$begingroup$
For each $n in mathbb N$, let $B_n=bigcup_{m=n}^infty A_m $. Clearly for each $n in mathbb N$ we have $B_n supseteq B_{n+1}$, and so ${B_n}_{n=1}^infty$ is a decreasing sequence of sets. Moreover, $P(B_1)<infty$ as a probability measure is finite. Therefore, $$Pleft(bigcap_{n=1}^infty B_nright)=lim_{nto infty} P(B_n)$$ by the continuity from above property of a measure which is $(3')$ here.
answered Dec 6 '18 at 3:38
Matt A PeltoMatt A Pelto
2,537620
$endgroup$
For each $n in mathbb N$, let $B_n=bigcup_{m=n}^infty A_m $. Clearly for each $n in mathbb N$ we have $B_n supseteq B_{n+1}$, and so ${B_n}_{n=1}^infty$ is a decreasing sequence of sets. Moreover, $P(B_1)<infty$ as a probability measure is finite. Therefore, $$Pleft(bigcap_{n=1}^infty B_nright)=lim_{nto infty} P(B_n)$$ by the continuity from above property of a measure which is $(3')$ here.
answered Dec 6 '18 at 3:38
Matt A PeltoMatt A Pelto
2,537620
edited Dec 6 '18 at 4:05
answered Dec 6 '18 at 3:38
Matt A PeltoMatt A Pelto
2,537620
answered Dec 6 '18 at 3:38
Matt A PeltoMatt A Pelto
2,537620
answered Dec 6 '18 at 3:38
Matt A PeltoMatt A Pelto
2,537620
2,537620
add a comment |
add a comment |
$begingroup$
Let $B_n:=bigcup_{mge n}A_m$. Then $B_{n+1}subseteq B_n$ and by continuity from above,
$$
mathsf{P}left(bigcap_{nge 1} B_nright)=lim_{ntoinfty} mathsf{P}(B_n).
$$
answered Dec 6 '18 at 3:32
d.k.o.d.k.o.
9,145628
$endgroup$
add a comment |
$begingroup$
Let $B_n:=bigcup_{mge n}A_m$. Then $B_{n+1}subseteq B_n$ and by continuity from above,
$$
mathsf{P}left(bigcap_{nge 1} B_nright)=lim_{ntoinfty} mathsf{P}(B_n).
$$
answered Dec 6 '18 at 3:32
d.k.o.d.k.o.
9,145628
$endgroup$
add a comment |
$begingroup$
Let $B_n:=bigcup_{mge n}A_m$. Then $B_{n+1}subseteq B_n$ and by continuity from above,
$$
mathsf{P}left(bigcap_{nge 1} B_nright)=lim_{ntoinfty} mathsf{P}(B_n).
$$
answered Dec 6 '18 at 3:32
d.k.o.d.k.o.
9,145628
$endgroup$
Let $B_n:=bigcup_{mge n}A_m$. Then $B_{n+1}subseteq B_n$ and by continuity from above,
$$
mathsf{P}left(bigcap_{nge 1} B_nright)=lim_{ntoinfty} mathsf{P}(B_n).
$$
answered Dec 6 '18 at 3:32
d.k.o.d.k.o.
9,145628
edited Dec 6 '18 at 4:00
answered Dec 6 '18 at 3:32
d.k.o.d.k.o.
9,145628
answered Dec 6 '18 at 3:32
d.k.o.d.k.o.
9,145628
answered Dec 6 '18 at 3:32
d.k.o.d.k.o.
9,145628
9,145628
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%2f3027990%2fis-it-true-p-cap-n-1-infty-cup-m-n-infty-a-m-lim-n-to-inftyp-cup%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
