Random measures by random fields

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Given a probability space $(Omega,mathcalA,mathbbP)$, we have a random field $X_t_t in T$, $Tsubset S_1times S_2$, for a measurable space $(S_1 times S_2,mathcalA_1timesmathcalA_2)$, i.e. $forall t in T$ $X_t:Omega to mathbbR$, $omega mapsto X_t(omega)$ is $mathcalA/mathcalB(mathbbR)$ measurable.



Now define $mu:Omegatimes S_1times S_2times mathbbR to mathbbR$ by
beginequation
mu(omegatimes s1,A_2,B)=sum_s_2 in A_2delta_X_(s1,s2)(omega)(B), B in mathcalB(mathbbR), A_2 in mathcalA_2.
endequation



Is $mu$ a random measure, i.e.



1) $mu(omegatimes s1,cdot,cdot)$ is a measure on $S_2timesmathbbR$,



2) $mu(cdot,A_2,B)$ is $mathcalAtimesmathcalA_1/mathcalB(mathbbR)$ measurable?



I think that $mu(omegatimes s1,cdot,cdot)$ should be a measure. However, I have no idea how I can ensure measurability, i.e. condition 2). Maybe someone has also an idea for a similar construction of $mu$.




stochastic-processes random-functions point-processes




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    Given a probability space $(Omega,mathcalA,mathbbP)$, we have a random field $X_t_t in T$, $Tsubset S_1times S_2$, for a measurable space $(S_1 times S_2,mathcalA_1timesmathcalA_2)$, i.e. $forall t in T$ $X_t:Omega to mathbbR$, $omega mapsto X_t(omega)$ is $mathcalA/mathcalB(mathbbR)$ measurable.



    Now define $mu:Omegatimes S_1times S_2times mathbbR to mathbbR$ by
    beginequation
    mu(omegatimes s1,A_2,B)=sum_s_2 in A_2delta_X_(s1,s2)(omega)(B), B in mathcalB(mathbbR), A_2 in mathcalA_2.
    endequation



    Is $mu$ a random measure, i.e.



    1) $mu(omegatimes s1,cdot,cdot)$ is a measure on $S_2timesmathbbR$,



    2) $mu(cdot,A_2,B)$ is $mathcalAtimesmathcalA_1/mathcalB(mathbbR)$ measurable?



    I think that $mu(omegatimes s1,cdot,cdot)$ should be a measure. However, I have no idea how I can ensure measurability, i.e. condition 2). Maybe someone has also an idea for a similar construction of $mu$.




    stochastic-processes random-functions point-processes




    share|cite|improve this question





















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      up vote
      1
      down vote

      favorite











      Given a probability space $(Omega,mathcalA,mathbbP)$, we have a random field $X_t_t in T$, $Tsubset S_1times S_2$, for a measurable space $(S_1 times S_2,mathcalA_1timesmathcalA_2)$, i.e. $forall t in T$ $X_t:Omega to mathbbR$, $omega mapsto X_t(omega)$ is $mathcalA/mathcalB(mathbbR)$ measurable.



      Now define $mu:Omegatimes S_1times S_2times mathbbR to mathbbR$ by
      beginequation
      mu(omegatimes s1,A_2,B)=sum_s_2 in A_2delta_X_(s1,s2)(omega)(B), B in mathcalB(mathbbR), A_2 in mathcalA_2.
      endequation



      Is $mu$ a random measure, i.e.



      1) $mu(omegatimes s1,cdot,cdot)$ is a measure on $S_2timesmathbbR$,



      2) $mu(cdot,A_2,B)$ is $mathcalAtimesmathcalA_1/mathcalB(mathbbR)$ measurable?



      I think that $mu(omegatimes s1,cdot,cdot)$ should be a measure. However, I have no idea how I can ensure measurability, i.e. condition 2). Maybe someone has also an idea for a similar construction of $mu$.




      stochastic-processes random-functions point-processes




      share|cite|improve this question











      Given a probability space $(Omega,mathcalA,mathbbP)$, we have a random field $X_t_t in T$, $Tsubset S_1times S_2$, for a measurable space $(S_1 times S_2,mathcalA_1timesmathcalA_2)$, i.e. $forall t in T$ $X_t:Omega to mathbbR$, $omega mapsto X_t(omega)$ is $mathcalA/mathcalB(mathbbR)$ measurable.



      Now define $mu:Omegatimes S_1times S_2times mathbbR to mathbbR$ by
      beginequation
      mu(omegatimes s1,A_2,B)=sum_s_2 in A_2delta_X_(s1,s2)(omega)(B), B in mathcalB(mathbbR), A_2 in mathcalA_2.
      endequation



      Is $mu$ a random measure, i.e.



      1) $mu(omegatimes s1,cdot,cdot)$ is a measure on $S_2timesmathbbR$,



      2) $mu(cdot,A_2,B)$ is $mathcalAtimesmathcalA_1/mathcalB(mathbbR)$ measurable?



      I think that $mu(omegatimes s1,cdot,cdot)$ should be a measure. However, I have no idea how I can ensure measurability, i.e. condition 2). Maybe someone has also an idea for a similar construction of $mu$.





      stochastic-processes random-functions point-processes





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      share|cite|improve this question




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      asked Jul 30 at 12:08









      Daniel Lingohr

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