$nu(A)=int mathcal{X}_{A}f(x)dmu$ measure associated functional $l_{nu}(g)=int g(x)f(x)dmu.$











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Let $mu$ be a positive Baire measure, $fin L^1(X,dmu)$ with $X$ compact hausdorff and $fgeq 0$.
For any Baire set $A$, define $nu(A)=int mathcal{X}_{A}f(x)dmu$.
Show that $nu$ is the one measure associated to the functional on $C(X)$,



$l_{nu}(g)=int g(x)f(x)dmu.$



Hello! In general, what are the steps to demonstrate that a measure comes from a functional one?










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    There is probably a theorem that gives you uniqueness of measures if the two measures coincide in a set of functions. And that would do it.
    – Will M.
    9 hours ago






  • 1




    You are supposed to show that $l_{nu} g=int g dnu$ for all $g in C(X)$. In other words you have to show that $int g(x)f(x)dmu(x)=int g(x)dnu(x)$.
    – Kavi Rama Murthy
    9 hours ago















up vote
0
down vote

favorite












Let $mu$ be a positive Baire measure, $fin L^1(X,dmu)$ with $X$ compact hausdorff and $fgeq 0$.
For any Baire set $A$, define $nu(A)=int mathcal{X}_{A}f(x)dmu$.
Show that $nu$ is the one measure associated to the functional on $C(X)$,



$l_{nu}(g)=int g(x)f(x)dmu.$



Hello! In general, what are the steps to demonstrate that a measure comes from a functional one?










share|cite|improve this question


















  • 1




    There is probably a theorem that gives you uniqueness of measures if the two measures coincide in a set of functions. And that would do it.
    – Will M.
    9 hours ago






  • 1




    You are supposed to show that $l_{nu} g=int g dnu$ for all $g in C(X)$. In other words you have to show that $int g(x)f(x)dmu(x)=int g(x)dnu(x)$.
    – Kavi Rama Murthy
    9 hours ago













up vote
0
down vote

favorite









up vote
0
down vote

favorite











Let $mu$ be a positive Baire measure, $fin L^1(X,dmu)$ with $X$ compact hausdorff and $fgeq 0$.
For any Baire set $A$, define $nu(A)=int mathcal{X}_{A}f(x)dmu$.
Show that $nu$ is the one measure associated to the functional on $C(X)$,



$l_{nu}(g)=int g(x)f(x)dmu.$



Hello! In general, what are the steps to demonstrate that a measure comes from a functional one?










share|cite|improve this question













Let $mu$ be a positive Baire measure, $fin L^1(X,dmu)$ with $X$ compact hausdorff and $fgeq 0$.
For any Baire set $A$, define $nu(A)=int mathcal{X}_{A}f(x)dmu$.
Show that $nu$ is the one measure associated to the functional on $C(X)$,



$l_{nu}(g)=int g(x)f(x)dmu.$



Hello! In general, what are the steps to demonstrate that a measure comes from a functional one?







functional-analysis measure-theory






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share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 9 hours ago









eraldcoil

21119




21119








  • 1




    There is probably a theorem that gives you uniqueness of measures if the two measures coincide in a set of functions. And that would do it.
    – Will M.
    9 hours ago






  • 1




    You are supposed to show that $l_{nu} g=int g dnu$ for all $g in C(X)$. In other words you have to show that $int g(x)f(x)dmu(x)=int g(x)dnu(x)$.
    – Kavi Rama Murthy
    9 hours ago














  • 1




    There is probably a theorem that gives you uniqueness of measures if the two measures coincide in a set of functions. And that would do it.
    – Will M.
    9 hours ago






  • 1




    You are supposed to show that $l_{nu} g=int g dnu$ for all $g in C(X)$. In other words you have to show that $int g(x)f(x)dmu(x)=int g(x)dnu(x)$.
    – Kavi Rama Murthy
    9 hours ago








1




1




There is probably a theorem that gives you uniqueness of measures if the two measures coincide in a set of functions. And that would do it.
– Will M.
9 hours ago




There is probably a theorem that gives you uniqueness of measures if the two measures coincide in a set of functions. And that would do it.
– Will M.
9 hours ago




1




1




You are supposed to show that $l_{nu} g=int g dnu$ for all $g in C(X)$. In other words you have to show that $int g(x)f(x)dmu(x)=int g(x)dnu(x)$.
– Kavi Rama Murthy
9 hours ago




You are supposed to show that $l_{nu} g=int g dnu$ for all $g in C(X)$. In other words you have to show that $int g(x)f(x)dmu(x)=int g(x)dnu(x)$.
– Kavi Rama Murthy
9 hours ago















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