Mixing
Understanding the definition of a measurable function
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I’m having trouble understanding the definition of a probability measurable function. The definition says that the preimage of events in the sigma algebra on the range must be an event in the sigma algebra on the domain. Why is this definition significant? What’s it trying to say?
stochastic-calculus
asked Jan 29 at 0:57
TeodorismTeodorism
314210
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add a comment |
$begingroup$
I’m having trouble understanding the definition of a probability measurable function. The definition says that the preimage of events in the sigma algebra on the range must be an event in the sigma algebra on the domain. Why is this definition significant? What’s it trying to say?
stochastic-calculus
asked Jan 29 at 0:57
TeodorismTeodorism
314210
$endgroup$
add a comment |
$begingroup$
I’m having trouble understanding the definition of a probability measurable function. The definition says that the preimage of events in the sigma algebra on the range must be an event in the sigma algebra on the domain. Why is this definition significant? What’s it trying to say?
stochastic-calculus
asked Jan 29 at 0:57
TeodorismTeodorism
314210
$endgroup$
I’m having trouble understanding the definition of a probability measurable function. The definition says that the preimage of events in the sigma algebra on the range must be an event in the sigma algebra on the domain. Why is this definition significant? What’s it trying to say?
stochastic-calculus
stochastic-calculus
asked Jan 29 at 0:57
TeodorismTeodorism
314210
asked Jan 29 at 0:57
TeodorismTeodorism
314210
asked Jan 29 at 0:57
TeodorismTeodorism
314210
asked Jan 29 at 0:57
TeodorismTeodorism
314210
asked Jan 29 at 0:57
TeodorismTeodorism
314210
314210
add a comment |
add a comment |
1 Answer
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Assume $X$ is a probability space, and $f:>Xto{mathbb R}$ is some scalar function. For a given number $ain{mathbb R}$ we want to be able to talk about the probability $Pbigl[f(x)leq abigr]$. This probability is equal to the measure of the set $f^{-1}({mathbb R}_{leq a})subset X$:$$Pbigl[f(x)leq abigr]=muleft( f^{-1}bigl({mathbb R}_{leq a})right) ,$$
and similarly
$$Pbigl[aleq f(x)leq b]=muleft( f^{-1}([a,b])right) .$$
Therefore we need the property that $f^{-1}$ of measurable sets in ${mathbb R}$ is measurable in $X$.
answered Jan 29 at 10:42
Christian BlatterChristian Blatter
176k8115327
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1 Answer
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1 Answer
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$begingroup$
Assume $X$ is a probability space, and $f:>Xto{mathbb R}$ is some scalar function. For a given number $ain{mathbb R}$ we want to be able to talk about the probability $Pbigl[f(x)leq abigr]$. This probability is equal to the measure of the set $f^{-1}({mathbb R}_{leq a})subset X$:$$Pbigl[f(x)leq abigr]=muleft( f^{-1}bigl({mathbb R}_{leq a})right) ,$$
and similarly
$$Pbigl[aleq f(x)leq b]=muleft( f^{-1}([a,b])right) .$$
Therefore we need the property that $f^{-1}$ of measurable sets in ${mathbb R}$ is measurable in $X$.
answered Jan 29 at 10:42
Christian BlatterChristian Blatter
176k8115327
$endgroup$
add a comment |
$begingroup$
Assume $X$ is a probability space, and $f:>Xto{mathbb R}$ is some scalar function. For a given number $ain{mathbb R}$ we want to be able to talk about the probability $Pbigl[f(x)leq abigr]$. This probability is equal to the measure of the set $f^{-1}({mathbb R}_{leq a})subset X$:$$Pbigl[f(x)leq abigr]=muleft( f^{-1}bigl({mathbb R}_{leq a})right) ,$$
and similarly
$$Pbigl[aleq f(x)leq b]=muleft( f^{-1}([a,b])right) .$$
Therefore we need the property that $f^{-1}$ of measurable sets in ${mathbb R}$ is measurable in $X$.
answered Jan 29 at 10:42
Christian BlatterChristian Blatter
176k8115327
$endgroup$
add a comment |
$begingroup$
Assume $X$ is a probability space, and $f:>Xto{mathbb R}$ is some scalar function. For a given number $ain{mathbb R}$ we want to be able to talk about the probability $Pbigl[f(x)leq abigr]$. This probability is equal to the measure of the set $f^{-1}({mathbb R}_{leq a})subset X$:$$Pbigl[f(x)leq abigr]=muleft( f^{-1}bigl({mathbb R}_{leq a})right) ,$$
and similarly
$$Pbigl[aleq f(x)leq b]=muleft( f^{-1}([a,b])right) .$$
Therefore we need the property that $f^{-1}$ of measurable sets in ${mathbb R}$ is measurable in $X$.
answered Jan 29 at 10:42
Christian BlatterChristian Blatter
176k8115327
$endgroup$
Assume $X$ is a probability space, and $f:>Xto{mathbb R}$ is some scalar function. For a given number $ain{mathbb R}$ we want to be able to talk about the probability $Pbigl[f(x)leq abigr]$. This probability is equal to the measure of the set $f^{-1}({mathbb R}_{leq a})subset X$:$$Pbigl[f(x)leq abigr]=muleft( f^{-1}bigl({mathbb R}_{leq a})right) ,$$
and similarly
$$Pbigl[aleq f(x)leq b]=muleft( f^{-1}([a,b])right) .$$
Therefore we need the property that $f^{-1}$ of measurable sets in ${mathbb R}$ is measurable in $X$.
answered Jan 29 at 10:42
Christian BlatterChristian Blatter
176k8115327
answered Jan 29 at 10:42
Christian BlatterChristian Blatter
176k8115327
answered Jan 29 at 10:42
Christian BlatterChristian Blatter
176k8115327
answered Jan 29 at 10:42
Christian BlatterChristian Blatter
176k8115327
176k8115327
add a comment |
add a comment |
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