First Variation of CDF inside an Indicator Function











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I would like to minimize the functional $mathcal{F}(mu) = int x I(F_mu(x)leqtau) dmu(x)$. However, I'm don't understand how to find the first variation of the term $I(F_mu(x)leqtau) = I(mu((-infty,x])leqtau)$, with respect to $mu$.



How do I compute the first variation $frac{delta }{delta mu}I(mu((-infty,x])leqtau)$?






measure-theory functional-equations gateaux-derivative







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    up vote
    1
    down vote

    favorite












    I would like to minimize the functional $mathcal{F}(mu) = int x I(F_mu(x)leqtau) dmu(x)$. However, I'm don't understand how to find the first variation of the term $I(F_mu(x)leqtau) = I(mu((-infty,x])leqtau)$, with respect to $mu$.



    How do I compute the first variation $frac{delta }{delta mu}I(mu((-infty,x])leqtau)$?






    measure-theory functional-equations gateaux-derivative







    share|cite|improve this question
























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      I would like to minimize the functional $mathcal{F}(mu) = int x I(F_mu(x)leqtau) dmu(x)$. However, I'm don't understand how to find the first variation of the term $I(F_mu(x)leqtau) = I(mu((-infty,x])leqtau)$, with respect to $mu$.



      How do I compute the first variation $frac{delta }{delta mu}I(mu((-infty,x])leqtau)$?






      measure-theory functional-equations gateaux-derivative







      share|cite|improve this question













      I would like to minimize the functional $mathcal{F}(mu) = int x I(F_mu(x)leqtau) dmu(x)$. However, I'm don't understand how to find the first variation of the term $I(F_mu(x)leqtau) = I(mu((-infty,x])leqtau)$, with respect to $mu$.



      How do I compute the first variation $frac{delta }{delta mu}I(mu((-infty,x])leqtau)$?






      measure-theory functional-equations gateaux-derivative




      measure-theory functional-equations gateaux-derivative






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Nov 18 at 0:09









      jdmartin86

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