What is the probability space in the Infinite Monkey Theorem?
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I tried searching everywhere but couldn't find a formalization of this.
I guess the set of possible outcomes $Omega$ is the set of all sequences of zeros and ones, where $1$ at the $n$th position means that the monkey typed Hamlet correctly at its $n$th attempt, and $0$ otherwise. The $sigma$-algebra $mathcal{M}$ on $Omega$ could be the one generated by the sets $A_n$, where $A_n$ is the set of all sequences with $1$ in the $n$th position.
How do we define a probability measure $mathbb{P}$ on $(Omega, mathcal{M})$ such that $mathbb{P}(A_n) = c > 0$, where $c$ is the probability of the monkey typing it correctly in one attempt?
probability-theory measure-theory
asked Jan 7 at 19:43
Doughnut PumpDoughnut Pump
481312
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add a comment |
$begingroup$
I tried searching everywhere but couldn't find a formalization of this.
I guess the set of possible outcomes $Omega$ is the set of all sequences of zeros and ones, where $1$ at the $n$th position means that the monkey typed Hamlet correctly at its $n$th attempt, and $0$ otherwise. The $sigma$-algebra $mathcal{M}$ on $Omega$ could be the one generated by the sets $A_n$, where $A_n$ is the set of all sequences with $1$ in the $n$th position.
How do we define a probability measure $mathbb{P}$ on $(Omega, mathcal{M})$ such that $mathbb{P}(A_n) = c > 0$, where $c$ is the probability of the monkey typing it correctly in one attempt?
probability-theory measure-theory
asked Jan 7 at 19:43
Doughnut PumpDoughnut Pump
481312
$endgroup$
$begingroup$
Are you sure that classically this probability measure is $sigma$-additive? You might want to look at some fair lotteries over infinite sets and/or to the probability of an infinite sequence of coin tosses. Both have been studied with a classic approach and also with nonstandard analysis. Some references are the papers Elementary numerosities and measures (logicandanalysis.org/index.php/jla/article/view/212/93) and Some applications of numerosities in measure theory (people.dm.unipi.it/dinasso/papers/Lincei2.pdf).
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– Emanuele Bottazzi
Jan 7 at 19:51
2
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This is simply the Borel $sigma$-algebra on $Omega = {0,1}^mathbb N$ with the product topology. It is the usual setting for considering sequences of Bernoulli random variables. The probability measure $mathbb P$ is the product of measures with $mathbb P({1})=c$, $mathbb P({0})=1-c$ on each component.
$endgroup$
– Robert Israel
Jan 7 at 19:57
add a comment |
$begingroup$
I tried searching everywhere but couldn't find a formalization of this.
I guess the set of possible outcomes $Omega$ is the set of all sequences of zeros and ones, where $1$ at the $n$th position means that the monkey typed Hamlet correctly at its $n$th attempt, and $0$ otherwise. The $sigma$-algebra $mathcal{M}$ on $Omega$ could be the one generated by the sets $A_n$, where $A_n$ is the set of all sequences with $1$ in the $n$th position.
How do we define a probability measure $mathbb{P}$ on $(Omega, mathcal{M})$ such that $mathbb{P}(A_n) = c > 0$, where $c$ is the probability of the monkey typing it correctly in one attempt?
probability-theory measure-theory
asked Jan 7 at 19:43
Doughnut PumpDoughnut Pump
481312
$endgroup$
I tried searching everywhere but couldn't find a formalization of this.
I guess the set of possible outcomes $Omega$ is the set of all sequences of zeros and ones, where $1$ at the $n$th position means that the monkey typed Hamlet correctly at its $n$th attempt, and $0$ otherwise. The $sigma$-algebra $mathcal{M}$ on $Omega$ could be the one generated by the sets $A_n$, where $A_n$ is the set of all sequences with $1$ in the $n$th position.
How do we define a probability measure $mathbb{P}$ on $(Omega, mathcal{M})$ such that $mathbb{P}(A_n) = c > 0$, where $c$ is the probability of the monkey typing it correctly in one attempt?
probability-theory measure-theory
probability-theory measure-theory
asked Jan 7 at 19:43
Doughnut PumpDoughnut Pump
481312
asked Jan 7 at 19:43
Doughnut PumpDoughnut Pump
481312
edited Jan 7 at 20:14
Doughnut Pump
asked Jan 7 at 19:43
Doughnut PumpDoughnut Pump
481312
asked Jan 7 at 19:43
Doughnut PumpDoughnut Pump
481312
asked Jan 7 at 19:43
Doughnut PumpDoughnut Pump
481312
481312
$begingroup$
Are you sure that classically this probability measure is $sigma$-additive? You might want to look at some fair lotteries over infinite sets and/or to the probability of an infinite sequence of coin tosses. Both have been studied with a classic approach and also with nonstandard analysis. Some references are the papers Elementary numerosities and measures (logicandanalysis.org/index.php/jla/article/view/212/93) and Some applications of numerosities in measure theory (people.dm.unipi.it/dinasso/papers/Lincei2.pdf).
$endgroup$
– Emanuele Bottazzi
Jan 7 at 19:51
2
$begingroup$
This is simply the Borel $sigma$-algebra on $Omega = {0,1}^mathbb N$ with the product topology. It is the usual setting for considering sequences of Bernoulli random variables. The probability measure $mathbb P$ is the product of measures with $mathbb P({1})=c$, $mathbb P({0})=1-c$ on each component.
$endgroup$
– Robert Israel
Jan 7 at 19:57
add a comment |
$begingroup$
Are you sure that classically this probability measure is $sigma$-additive? You might want to look at some fair lotteries over infinite sets and/or to the probability of an infinite sequence of coin tosses. Both have been studied with a classic approach and also with nonstandard analysis. Some references are the papers Elementary numerosities and measures (logicandanalysis.org/index.php/jla/article/view/212/93) and Some applications of numerosities in measure theory (people.dm.unipi.it/dinasso/papers/Lincei2.pdf).
$endgroup$
– Emanuele Bottazzi
Jan 7 at 19:51
2
$begingroup$
This is simply the Borel $sigma$-algebra on $Omega = {0,1}^mathbb N$ with the product topology. It is the usual setting for considering sequences of Bernoulli random variables. The probability measure $mathbb P$ is the product of measures with $mathbb P({1})=c$, $mathbb P({0})=1-c$ on each component.
$endgroup$
– Robert Israel
Jan 7 at 19:57
$begingroup$
Are you sure that classically this probability measure is $sigma$-additive? You might want to look at some fair lotteries over infinite sets and/or to the probability of an infinite sequence of coin tosses. Both have been studied with a classic approach and also with nonstandard analysis. Some references are the papers Elementary numerosities and measures (logicandanalysis.org/index.php/jla/article/view/212/93) and Some applications of numerosities in measure theory (people.dm.unipi.it/dinasso/papers/Lincei2.pdf).
$endgroup$
– Emanuele Bottazzi
Jan 7 at 19:51
$begingroup$
Are you sure that classically this probability measure is $sigma$-additive? You might want to look at some fair lotteries over infinite sets and/or to the probability of an infinite sequence of coin tosses. Both have been studied with a classic approach and also with nonstandard analysis. Some references are the papers Elementary numerosities and measures (logicandanalysis.org/index.php/jla/article/view/212/93) and Some applications of numerosities in measure theory (people.dm.unipi.it/dinasso/papers/Lincei2.pdf).
$endgroup$
– Emanuele Bottazzi
Jan 7 at 19:51
2
2
$begingroup$
This is simply the Borel $sigma$-algebra on $Omega = {0,1}^mathbb N$ with the product topology. It is the usual setting for considering sequences of Bernoulli random variables. The probability measure $mathbb P$ is the product of measures with $mathbb P({1})=c$, $mathbb P({0})=1-c$ on each component.
$endgroup$
– Robert Israel
Jan 7 at 19:57
$begingroup$
This is simply the Borel $sigma$-algebra on $Omega = {0,1}^mathbb N$ with the product topology. It is the usual setting for considering sequences of Bernoulli random variables. The probability measure $mathbb P$ is the product of measures with $mathbb P({1})=c$, $mathbb P({0})=1-c$ on each component.
$endgroup$
– Robert Israel
Jan 7 at 19:57
add a comment |
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Are you sure that classically this probability measure is $sigma$-additive? You might want to look at some fair lotteries over infinite sets and/or to the probability of an infinite sequence of coin tosses. Both have been studied with a classic approach and also with nonstandard analysis. Some references are the papers Elementary numerosities and measures (logicandanalysis.org/index.php/jla/article/view/212/93) and Some applications of numerosities in measure theory (people.dm.unipi.it/dinasso/papers/Lincei2.pdf).
$endgroup$
– Emanuele Bottazzi
Jan 7 at 19:51
2
$begingroup$
This is simply the Borel $sigma$-algebra on $Omega = {0,1}^mathbb N$ with the product topology. It is the usual setting for considering sequences of Bernoulli random variables. The probability measure $mathbb P$ is the product of measures with $mathbb P({1})=c$, $mathbb P({0})=1-c$ on each component.
$endgroup$
– Robert Israel
Jan 7 at 19:57