Proving outer measure is additive over disjoint open cells (without Caratheodory)
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Is there a way to prove that outer measure is additive over disjoint open cells without Caratheodory's condition/theorem?
Ie, if the $E_k$ are mutually disjoint open cells, I want to prove that $m^*(cup E_k) = Sigma V(E_k)$. The <= direction is trivial through the infimum definition of outer measure, but I'm not sure about the >= direction.
For all $epsilon>0$, we know there exists an open covering $A_k$ such that $m^*(cup E_k) + epsilon >= Sigma V(A_k)$, but this isn't true for all open covers, so I'm not sure how to relate this to the $E_k$ cover.
measure-theory lebesgue-measure outer-measure
asked Dec 19 '18 at 19:12
user49404user49404
578
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add a comment |
$begingroup$
Is there a way to prove that outer measure is additive over disjoint open cells without Caratheodory's condition/theorem?
Ie, if the $E_k$ are mutually disjoint open cells, I want to prove that $m^*(cup E_k) = Sigma V(E_k)$. The <= direction is trivial through the infimum definition of outer measure, but I'm not sure about the >= direction.
For all $epsilon>0$, we know there exists an open covering $A_k$ such that $m^*(cup E_k) + epsilon >= Sigma V(A_k)$, but this isn't true for all open covers, so I'm not sure how to relate this to the $E_k$ cover.
measure-theory lebesgue-measure outer-measure
asked Dec 19 '18 at 19:12
user49404user49404
578
$endgroup$
add a comment |
$begingroup$
Is there a way to prove that outer measure is additive over disjoint open cells without Caratheodory's condition/theorem?
Ie, if the $E_k$ are mutually disjoint open cells, I want to prove that $m^*(cup E_k) = Sigma V(E_k)$. The <= direction is trivial through the infimum definition of outer measure, but I'm not sure about the >= direction.
For all $epsilon>0$, we know there exists an open covering $A_k$ such that $m^*(cup E_k) + epsilon >= Sigma V(A_k)$, but this isn't true for all open covers, so I'm not sure how to relate this to the $E_k$ cover.
measure-theory lebesgue-measure outer-measure
asked Dec 19 '18 at 19:12
user49404user49404
578
$endgroup$
Is there a way to prove that outer measure is additive over disjoint open cells without Caratheodory's condition/theorem?
Ie, if the $E_k$ are mutually disjoint open cells, I want to prove that $m^*(cup E_k) = Sigma V(E_k)$. The <= direction is trivial through the infimum definition of outer measure, but I'm not sure about the >= direction.
For all $epsilon>0$, we know there exists an open covering $A_k$ such that $m^*(cup E_k) + epsilon >= Sigma V(A_k)$, but this isn't true for all open covers, so I'm not sure how to relate this to the $E_k$ cover.
measure-theory lebesgue-measure outer-measure
measure-theory lebesgue-measure outer-measure
asked Dec 19 '18 at 19:12
user49404user49404
578
asked Dec 19 '18 at 19:12
user49404user49404
578
asked Dec 19 '18 at 19:12
user49404user49404
578
asked Dec 19 '18 at 19:12
user49404user49404
578
asked Dec 19 '18 at 19:12
user49404user49404
578
578
add a comment |
add a comment |
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