Mixing
Open set with compact closure has finite Radon premeasure
$begingroup$
Let $X$ be a locally compact Hausdorff space and $I$ a positive linear functional on $C_{c}(X)$.
For each nonempty open subset $O$ of $X$, define
$μ(O) = sup${$ I(f): f∈C_{c}(X), 0≤f≤1, supp f ⊆ O$}.
How to show that if an open set $O$ has a compact closure, then $μ(O)<∞ $? I found the proof in the 4th edition of Royden on p.460 a bit confusing. Could someone please help me? Thank you very much.
real-analysis functional-analysis riesz-representation-theorem
asked Feb 3 at 3:36
DeepleeqeDeepleeqe
1409
$endgroup$
add a comment |
$begingroup$
Let $X$ be a locally compact Hausdorff space and $I$ a positive linear functional on $C_{c}(X)$.
For each nonempty open subset $O$ of $X$, define
$μ(O) = sup${$ I(f): f∈C_{c}(X), 0≤f≤1, supp f ⊆ O$}.
How to show that if an open set $O$ has a compact closure, then $μ(O)<∞ $? I found the proof in the 4th edition of Royden on p.460 a bit confusing. Could someone please help me? Thank you very much.
real-analysis functional-analysis riesz-representation-theorem
asked Feb 3 at 3:36
DeepleeqeDeepleeqe
1409
$endgroup$
add a comment |
$begingroup$
Let $X$ be a locally compact Hausdorff space and $I$ a positive linear functional on $C_{c}(X)$.
For each nonempty open subset $O$ of $X$, define
$μ(O) = sup${$ I(f): f∈C_{c}(X), 0≤f≤1, supp f ⊆ O$}.
How to show that if an open set $O$ has a compact closure, then $μ(O)<∞ $? I found the proof in the 4th edition of Royden on p.460 a bit confusing. Could someone please help me? Thank you very much.
real-analysis functional-analysis riesz-representation-theorem
asked Feb 3 at 3:36
DeepleeqeDeepleeqe
1409
$endgroup$
Let $X$ be a locally compact Hausdorff space and $I$ a positive linear functional on $C_{c}(X)$.
For each nonempty open subset $O$ of $X$, define
$μ(O) = sup${$ I(f): f∈C_{c}(X), 0≤f≤1, supp f ⊆ O$}.
How to show that if an open set $O$ has a compact closure, then $μ(O)<∞ $? I found the proof in the 4th edition of Royden on p.460 a bit confusing. Could someone please help me? Thank you very much.
real-analysis functional-analysis riesz-representation-theorem
real-analysis functional-analysis riesz-representation-theorem
asked Feb 3 at 3:36
DeepleeqeDeepleeqe
1409
asked Feb 3 at 3:36
DeepleeqeDeepleeqe
1409
edited Feb 3 at 3:44
Deepleeqe
asked Feb 3 at 3:36
DeepleeqeDeepleeqe
1409
asked Feb 3 at 3:36
DeepleeqeDeepleeqe
1409
asked Feb 3 at 3:36
DeepleeqeDeepleeqe
1409
1409
add a comment |
add a comment |
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