Mixing
Extension of $sigma$-algebra with Dirac-measure
$begingroup$
Let $(X, xi, mu)$ be a measure space where $mu := delta_x$ is the Dirac-Measure for a fixed $x in X$. I want to find a more explicit description for the extended $sigma$-algebra:
$$xi_mu := {A in mathcal{P}(X) : exists B, C in xi: A ,triangle ,B subseteq C text{ and } mu(C) = 0}$$
My current approach looks like this: For $A in mathcal{P}(X)$:
$$A in xi_{mu} Leftrightarrow exists B, C in xi: A ,triangle ,B subseteq C text{ and } mu(C) = 0 \ Leftrightarrow exists B, C in xi: A ,triangle ,B subseteq C text{ and } x notin C \ Rightarrow exists B in xi: x notin A ,triangle, B$$
but I'm not sure if this leads to anything more explicit. How would you go from here?
measure-theory
asked Jan 7 at 18:59
user7802048user7802048
335211
$endgroup$
add a comment |
$begingroup$
Let $(X, xi, mu)$ be a measure space where $mu := delta_x$ is the Dirac-Measure for a fixed $x in X$. I want to find a more explicit description for the extended $sigma$-algebra:
$$xi_mu := {A in mathcal{P}(X) : exists B, C in xi: A ,triangle ,B subseteq C text{ and } mu(C) = 0}$$
My current approach looks like this: For $A in mathcal{P}(X)$:
$$A in xi_{mu} Leftrightarrow exists B, C in xi: A ,triangle ,B subseteq C text{ and } mu(C) = 0 \ Leftrightarrow exists B, C in xi: A ,triangle ,B subseteq C text{ and } x notin C \ Rightarrow exists B in xi: x notin A ,triangle, B$$
but I'm not sure if this leads to anything more explicit. How would you go from here?
measure-theory
asked Jan 7 at 18:59
user7802048user7802048
335211
$endgroup$
$begingroup$
You can do casework on whether $x$ is in $A$, if you want. That is, if $x in A$, then you want some $B in xi$ with $x in B$. And if $x not in A$, then you want some $B in xi$ with $x not in B$.
$endgroup$
– mathworker21
Jan 7 at 19:12
add a comment |
$begingroup$
Let $(X, xi, mu)$ be a measure space where $mu := delta_x$ is the Dirac-Measure for a fixed $x in X$. I want to find a more explicit description for the extended $sigma$-algebra:
$$xi_mu := {A in mathcal{P}(X) : exists B, C in xi: A ,triangle ,B subseteq C text{ and } mu(C) = 0}$$
My current approach looks like this: For $A in mathcal{P}(X)$:
$$A in xi_{mu} Leftrightarrow exists B, C in xi: A ,triangle ,B subseteq C text{ and } mu(C) = 0 \ Leftrightarrow exists B, C in xi: A ,triangle ,B subseteq C text{ and } x notin C \ Rightarrow exists B in xi: x notin A ,triangle, B$$
but I'm not sure if this leads to anything more explicit. How would you go from here?
measure-theory
asked Jan 7 at 18:59
user7802048user7802048
335211
$endgroup$
Let $(X, xi, mu)$ be a measure space where $mu := delta_x$ is the Dirac-Measure for a fixed $x in X$. I want to find a more explicit description for the extended $sigma$-algebra:
$$xi_mu := {A in mathcal{P}(X) : exists B, C in xi: A ,triangle ,B subseteq C text{ and } mu(C) = 0}$$
My current approach looks like this: For $A in mathcal{P}(X)$:
$$A in xi_{mu} Leftrightarrow exists B, C in xi: A ,triangle ,B subseteq C text{ and } mu(C) = 0 \ Leftrightarrow exists B, C in xi: A ,triangle ,B subseteq C text{ and } x notin C \ Rightarrow exists B in xi: x notin A ,triangle, B$$
but I'm not sure if this leads to anything more explicit. How would you go from here?
measure-theory
measure-theory
asked Jan 7 at 18:59
user7802048user7802048
335211
asked Jan 7 at 18:59
user7802048user7802048
335211
asked Jan 7 at 18:59
user7802048user7802048
335211
asked Jan 7 at 18:59
user7802048user7802048
335211
asked Jan 7 at 18:59
user7802048user7802048
335211
335211
$begingroup$
You can do casework on whether $x$ is in $A$, if you want. That is, if $x in A$, then you want some $B in xi$ with $x in B$. And if $x not in A$, then you want some $B in xi$ with $x not in B$.
$endgroup$
– mathworker21
Jan 7 at 19:12
add a comment |
$begingroup$
You can do casework on whether $x$ is in $A$, if you want. That is, if $x in A$, then you want some $B in xi$ with $x in B$. And if $x not in A$, then you want some $B in xi$ with $x not in B$.
$endgroup$
– mathworker21
Jan 7 at 19:12
$begingroup$
You can do casework on whether $x$ is in $A$, if you want. That is, if $x in A$, then you want some $B in xi$ with $x in B$. And if $x not in A$, then you want some $B in xi$ with $x not in B$.
$endgroup$
– mathworker21
Jan 7 at 19:12
$begingroup$
You can do casework on whether $x$ is in $A$, if you want. That is, if $x in A$, then you want some $B in xi$ with $x in B$. And if $x not in A$, then you want some $B in xi$ with $x not in B$.
$endgroup$
– mathworker21
Jan 7 at 19:12
add a comment |
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$begingroup$
You can do casework on whether $x$ is in $A$, if you want. That is, if $x in A$, then you want some $B in xi$ with $x in B$. And if $x not in A$, then you want some $B in xi$ with $x not in B$.
$endgroup$
– mathworker21
Jan 7 at 19:12