The Wiener measure is (in the classical sense) a Gaussian measure on the Banach space $C[0,1]:=\{f:[0,1] \to \mathbb{R} \mid f\text{ is continuous and } f(0)=1\}$.


The Wiener process is a stochastic process whose definition can be found in any textbook. In any text, the stochastic integrals or stochastic differential equations use the notation $dW$ frequently, which should denote the Wiener measure, I suppose.


However, I cannot figure out the exact relation between the Wiener 'process' and 'measure'. Wikipedia says that the Wiener process induces the Wiener measure but what exactly does that mean?


I am afraid this question might not belong to MO but I ask here. Could anyone please clarify and help me understand?



What exactly is the relation between the Wiener process and Wiener measure?
The Wiener measure $w$ is the distribution of the Wiener process/random function $W$ on $C[0,1]$; that is,
$$P(W\in A)=w(A)$$
for all Borel sets $A\subseteq C[0,1]$. Here "Borel sets" can be replaced by "open sets" or "closed sets".


Equivalently,
$$Ef(W)=\int\_{C[0,1]}f\,dw$$
for all (say) nonnegative Borel-measurable functions $f\colon C[0,1]\to\mathbb R$. Here "nonnegative Borel-measurable" can be replaced by (say) "bounded continuous".


