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Let $(\Omega,\mathbb{F},P)$ be a probability space and $E$ be an infinite dimensional Banach space and $\mathbb{B}$ be the $\sigma$-algebra of Borel subset of $E$.

Let $X$ be random function defined on $(\Omega,\mathbb{F},P)$ taking values in $(E,\mathbb{B})$. Example of such random function $X$ is stochastic processes with square integrable sample paths on the real line when $E= L_2(R)$.

Let $P_X$ be the induced probability measure induced by by $X$ on $(E,\mathbb{B})$.

I want to define the density of $X$ using Radon-Nikodym Theorem. So I need a $\sigma$-finite measure $\mu$ (say) which dominates the $\sigma$-finite measure $P_X$.

I am stuck here. How to find $\mu$ in this case?

Any suggestion, solution or useful reference will be gratefully appreciated.

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  • $\begingroup$ @ janak : I think that what you want to say is that $P_X$ is the push forward measure of $P$ by $X$. Best regards $\endgroup$
    – The Bridge
    Commented Nov 28, 2014 at 10:33
  • $\begingroup$ @TheBridge: Yes, $P_X$ is push forward measure. $\endgroup$
    – Janak
    Commented Nov 28, 2014 at 10:37
  • $\begingroup$ To stimulate this thread you should try to say more precisely what you are looking for. As stated, your question does not make much sense (you could take $\mu=P_X$ and density $1$). For $E=C([0,\infty))$ the Wiener measure (distribution of Brownian motion) is certainly an interesting measure on $E$, but without more context it is not clear whether this is an interesting case for you. $\endgroup$
    – Jochen Wengenroth
    Commented Dec 1, 2014 at 16:04
  • $\begingroup$ The context of my question is: I have a set of parametric curves $\beta(t)=(x(t),y(t))', t\in [0,1]$ (for example $x(t)=sin t$ and $y(t)=cos t$). I consider these curves are random sample from population of parametric curves and I try to understand whether I can define probability density function of this random curves. $\endgroup$
    – Janak
    Commented Dec 1, 2014 at 18:32
  • $\begingroup$ I understand the space of such parametric curves is no longer a linear space. But I think it could be a valid question to ask whether the density of the random curve are exists or not. $\endgroup$
    – Janak
    Commented Dec 1, 2014 at 18:46

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