Let $(W\_t)\_{t\ge0}$ be a continuous Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$ and $$\pi\_t:C([0,\infty))\to\mathbb R\;,\;\;\;x\mapsto x(t)$$ for $t\ge0$. Equip $C([0,\infty))$ with the topology of compact convergence. Then $$\mathcal B(C([0,\infty))=\sigma\left(\pi\_t,t\ge0\right)\tag1$$ and hence $W:\Omega\to C([0,\infty))$ is $\left(\mathcal A,\mathcal B(\tau)\right)$-measurable.

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> So, the process $(\pi\_t)\_{t\ge0}$ is a Brownian motion on $(C[0,\infty),\mathcal B(C[0,\infty)),\operatorname P\_0)$, where $\operatorname P\_0:=\operatorname P\circ\:W^{-1}$.
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> Now it is tempting to consider for every $x\in\mathbb R$ a separate continuous Brownian motion *started in $x$* and obtain probability measure $\operatorname P\_x$ on $(C[0,\infty),\mathcal B(C[0,\infty))$ in this way.
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> **Question**: Are we able to do this such that the map $x\mapsto\operatorname P\_x(B)$ is Borel measurable for all $B\in\mathcal B(C([0,\infty))$?
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