Kolmogorov extension theorem for measures on the space of continuous functions












2














In a text I am reading occurs the following:



One has a family of probability measures $(P_T)_{Tgeq 0}$ such that $P_T$ is a measure on the measurable space $(C([0,T],Bbb R^d), mathcal B (C([0,T],Bbb R^d)) )$.



It is stated that consistency of the family $(P_T)_{Tgeq 0}$ is yielding a measure $P$ on $C([0,infty),Bbb R^d)$.




  1. I think consistency means if we take $pi^S_T: C([0,S],Bbb R^d) to C([0,T],Bbb R^d)$,$fmapsto fvert_{[0,T]}$, then for every $Sgeq T$ we have $P_T = P_S circ (pi_T^S)^{-1}$.


  2. P seems to satisfy $P_T = Pcirc(pi_T)^{-1}$, where $pi_T: C([0,infty),Bbb R^d) to C([0,T],Bbb R^d)$,$fmapsto fvert_{[0,T]}$



Is this a trivial fact? My first thought was to apply the Kolmogorov extension theorem but I did not come far enough with it.






probability-theory measure-theory stochastic-processes







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    2














    In a text I am reading occurs the following:



    One has a family of probability measures $(P_T)_{Tgeq 0}$ such that $P_T$ is a measure on the measurable space $(C([0,T],Bbb R^d), mathcal B (C([0,T],Bbb R^d)) )$.



    It is stated that consistency of the family $(P_T)_{Tgeq 0}$ is yielding a measure $P$ on $C([0,infty),Bbb R^d)$.




    1. I think consistency means if we take $pi^S_T: C([0,S],Bbb R^d) to C([0,T],Bbb R^d)$,$fmapsto fvert_{[0,T]}$, then for every $Sgeq T$ we have $P_T = P_S circ (pi_T^S)^{-1}$.


    2. P seems to satisfy $P_T = Pcirc(pi_T)^{-1}$, where $pi_T: C([0,infty),Bbb R^d) to C([0,T],Bbb R^d)$,$fmapsto fvert_{[0,T]}$



    Is this a trivial fact? My first thought was to apply the Kolmogorov extension theorem but I did not come far enough with it.






    probability-theory measure-theory stochastic-processes







    share|cite|improve this question



























      2












      2








      2


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      In a text I am reading occurs the following:



      One has a family of probability measures $(P_T)_{Tgeq 0}$ such that $P_T$ is a measure on the measurable space $(C([0,T],Bbb R^d), mathcal B (C([0,T],Bbb R^d)) )$.



      It is stated that consistency of the family $(P_T)_{Tgeq 0}$ is yielding a measure $P$ on $C([0,infty),Bbb R^d)$.




      1. I think consistency means if we take $pi^S_T: C([0,S],Bbb R^d) to C([0,T],Bbb R^d)$,$fmapsto fvert_{[0,T]}$, then for every $Sgeq T$ we have $P_T = P_S circ (pi_T^S)^{-1}$.


      2. P seems to satisfy $P_T = Pcirc(pi_T)^{-1}$, where $pi_T: C([0,infty),Bbb R^d) to C([0,T],Bbb R^d)$,$fmapsto fvert_{[0,T]}$



      Is this a trivial fact? My first thought was to apply the Kolmogorov extension theorem but I did not come far enough with it.






      probability-theory measure-theory stochastic-processes







      share|cite|improve this question















      In a text I am reading occurs the following:



      One has a family of probability measures $(P_T)_{Tgeq 0}$ such that $P_T$ is a measure on the measurable space $(C([0,T],Bbb R^d), mathcal B (C([0,T],Bbb R^d)) )$.



      It is stated that consistency of the family $(P_T)_{Tgeq 0}$ is yielding a measure $P$ on $C([0,infty),Bbb R^d)$.




      1. I think consistency means if we take $pi^S_T: C([0,S],Bbb R^d) to C([0,T],Bbb R^d)$,$fmapsto fvert_{[0,T]}$, then for every $Sgeq T$ we have $P_T = P_S circ (pi_T^S)^{-1}$.


      2. P seems to satisfy $P_T = Pcirc(pi_T)^{-1}$, where $pi_T: C([0,infty),Bbb R^d) to C([0,T],Bbb R^d)$,$fmapsto fvert_{[0,T]}$



      Is this a trivial fact? My first thought was to apply the Kolmogorov extension theorem but I did not come far enough with it.






      probability-theory measure-theory stochastic-processes




      probability-theory measure-theory stochastic-processes






      share|cite|improve this question















      share|cite|improve this question













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      edited Jan 4 at 16:18







      Falrach

















      asked Jan 4 at 15:11









      FalrachFalrach

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