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YCor
YCor
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Is Completioncompletion of Measures Equivalentmeasures equivalent to Completioncompletion of Sigma Algebrassigma algebras as Metric Spaces With Respectmetric spaces with respect to Measuresmeasures?

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Michael Hardy
Michael Hardy
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An alternative way to get the Lebesgue $\sigma $-algebra $\mathcal{L} $ from the Borel algebra $B$ is to set $E\sim J$ iff $d(E,J):=\lambda(E\Delta J)=0$$d(E,J):=\lambda(E\mathbin\Delta J)=0$ for $E,J\in B$. Then the completion of $(X,d)$ as a metric space, where $X=B/\sim$$X=B/{\sim}$, is equivalent to completion of $B$.

My question is: Are there known generalizations? Given a measure space $(X,\Sigma ,\mu)$, what are the conditions on $X$ and $\Sigma$ so that the completion of the measure space is equivalent to the completion of $M:=\Sigma/\sim$$M := \Sigma/{\sim}$ as a metric space, with $\sim $ as above? Do we have to use the symmetric difference to define $d$ on $M$?

An alternative way to get the Lebesgue $\sigma $-algebra $\mathcal{L} $ from the Borel algebra $B$ is to set $E\sim J$ iff $d(E,J):=\lambda(E\Delta J)=0$ for $E,J\in B$. Then the completion of $(X,d)$ as a metric space, where $X=B/\sim$, is equivalent to completion of $B$.

My question is: Are there known generalizations? Given a measure space $(X,\Sigma ,\mu)$, what are the conditions on $X$ and $\Sigma$ so that the completion of the measure space is equivalent to the completion of $M:=\Sigma/\sim$ as a metric space, with $\sim $ as above? Do we have to use the symmetric difference to define $d$ on $M$?

An alternative way to get the Lebesgue $\sigma $-algebra $\mathcal{L} $ from the Borel algebra $B$ is to set $E\sim J$ iff $d(E,J):=\lambda(E\mathbin\Delta J)=0$ for $E,J\in B$. Then the completion of $(X,d)$ as a metric space, where $X=B/{\sim}$, is equivalent to completion of $B$.

My question is: Are there known generalizations? Given a measure space $(X,\Sigma ,\mu)$, what are the conditions on $X$ and $\Sigma$ so that the completion of the measure space is equivalent to the completion of $M := \Sigma/{\sim}$ as a metric space, with $\sim $ as above? Do we have to use the symmetric difference to define $d$ on $M$?

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DJ Forklift
DJ Forklift
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Is Completion of Measures Equivalent to Completion of Sigma Algebras as Metric Spaces With Respect to Measures

An alternative way to get the Lebesgue $\sigma $-algebra $\mathcal{L} $ from the Borel algebra $B$ is to set $E\sim J$ iff $d(E,J):=\lambda(E\Delta J)=0$ for $E,J\in B$. Then the completion of $(X,d)$ as a metric space, where $X=B/\sim$, is equivalent to completion of $B$.

My question is: Are there known generalizations? Given a measure space $(X,\Sigma ,\mu)$, what are the conditions on $X$ and $\Sigma$ so that the completion of the measure space is equivalent to the completion of $M:=\Sigma/\sim$ as a metric space, with $\sim $ as above? Do we have to use the symmetric difference to define $d$ on $M$?