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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 $X:[0,1]\rightarrow \mathbb{R}$, 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.

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.

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 $X:[0,1]\rightarrow \mathbb{R}$, be random function defined on $(\Omega,\mathbb{F},P)$ taking values in $(E,\mathbb{B})$. Examples of such random function $X$ are stochastic processes with continuous sample paths on a finite interval $[0,1]$ with $E=C[0,1]$ associated with supremum norm and 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.

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 $X:[0,1]\rightarrow \mathbb{R}$, 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.

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 $X:[0,1]\rightarrow \mathbb{R}$, be random function defined on $(\Omega,\mathbb{F},P)$ taking values in $(E,\mathbb{B})$. Let $P_X$ be the induced probability measure 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.

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 $X:[0,1]\rightarrow \mathbb{R}$, be random function defined on $(\Omega,\mathbb{F},P)$ taking values in $(E,\mathbb{B})$. Examples of such random function $X$ are stochastic processes with continuous sample paths on a finite interval $[0,1]$ with $E=C[0,1]$ associated with supremum norm and 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.