The Wiener measure $w$ is the distribution of the Wiener process $W$ 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 $f\colon C[0,1]\to\mathbb R$.

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".