Integrating with respect to surface measure

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How does one integrate with respect to a surface measure, is it the same as a surface integral? I've tried a google search but couldn't find much.



In particular I have the following problem but can't begin as i don't know the above!



Let $Gamma in L^1(S^n-1)$ where $S^n-1$ is the (n-1) dimensional sphere on $mathbbR^n.$ Given $z in mathbbR^n backslash 0$ the projection onto $S^n-1$ is given by:
$$ z' = fracz in S^n-1$$



We define the operator $$Tf(x) = lim_epsilon to 0int_ > epsilonfracGamma(z')^nf(x-z)dz.$$



Show that the limit above exists for $f in mathcalS(mathbbR^n)$ i.e. a Schwartz function by assuming that $int_S^n-1Gamma(z')d sigma(z')=0$ where $d sigma$ denotes the surface measure on $S^n-1$.




real-analysis harmonic-analysis surface-integrals




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    up vote
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    down vote

    favorite












    How does one integrate with respect to a surface measure, is it the same as a surface integral? I've tried a google search but couldn't find much.



    In particular I have the following problem but can't begin as i don't know the above!



    Let $Gamma in L^1(S^n-1)$ where $S^n-1$ is the (n-1) dimensional sphere on $mathbbR^n.$ Given $z in mathbbR^n backslash 0$ the projection onto $S^n-1$ is given by:
    $$ z' = fracz in S^n-1$$



    We define the operator $$Tf(x) = lim_epsilon to 0int_ > epsilonfracGamma(z')^nf(x-z)dz.$$



    Show that the limit above exists for $f in mathcalS(mathbbR^n)$ i.e. a Schwartz function by assuming that $int_S^n-1Gamma(z')d sigma(z')=0$ where $d sigma$ denotes the surface measure on $S^n-1$.




    real-analysis harmonic-analysis surface-integrals




    share|cite|improve this question





















      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      How does one integrate with respect to a surface measure, is it the same as a surface integral? I've tried a google search but couldn't find much.



      In particular I have the following problem but can't begin as i don't know the above!



      Let $Gamma in L^1(S^n-1)$ where $S^n-1$ is the (n-1) dimensional sphere on $mathbbR^n.$ Given $z in mathbbR^n backslash 0$ the projection onto $S^n-1$ is given by:
      $$ z' = fracz in S^n-1$$



      We define the operator $$Tf(x) = lim_epsilon to 0int_ > epsilonfracGamma(z')^nf(x-z)dz.$$



      Show that the limit above exists for $f in mathcalS(mathbbR^n)$ i.e. a Schwartz function by assuming that $int_S^n-1Gamma(z')d sigma(z')=0$ where $d sigma$ denotes the surface measure on $S^n-1$.




      real-analysis harmonic-analysis surface-integrals




      share|cite|improve this question











      How does one integrate with respect to a surface measure, is it the same as a surface integral? I've tried a google search but couldn't find much.



      In particular I have the following problem but can't begin as i don't know the above!



      Let $Gamma in L^1(S^n-1)$ where $S^n-1$ is the (n-1) dimensional sphere on $mathbbR^n.$ Given $z in mathbbR^n backslash 0$ the projection onto $S^n-1$ is given by:
      $$ z' = fracz in S^n-1$$



      We define the operator $$Tf(x) = lim_epsilon to 0int_ > epsilonfracGamma(z')^nf(x-z)dz.$$



      Show that the limit above exists for $f in mathcalS(mathbbR^n)$ i.e. a Schwartz function by assuming that $int_S^n-1Gamma(z')d sigma(z')=0$ where $d sigma$ denotes the surface measure on $S^n-1$.





      real-analysis harmonic-analysis surface-integrals





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