Brownian scaling

By [F2], $a=1\mathrm{/}\sqrt{c}$ is a well-defined positive real and $a^{2}c=1$. For any finite times $t_1,\dots ,t_n$ and coefficients $u_1,\dots ,u_n$, ${\sum }_{j=1}^n{u}_j{Y}_{t_j}=a{\sum }_{j=1}^n{u}_j{B}_{c{t}_j}\mathrm{.}$ The right side is normal by the Gaussian characterization in [F1], including when coefficients vanish or times repeat. Hence $Y$ is Gaussian and centered.

For $s,t\ge 0$, $\mathrm{Cov}(Y_s,Y_t)=a^2\mathrm{Cov}(B_{cs},B_{ct})=a^2\min (cs,ct)=\min (s,t)\mathrm{.}$ Also $Y_0=a{B}_0=0$ almost surely. By [F1], $Y$ therefore has mutually independent increments $Y_{t_j}-Y_{t_{j-1}}\sim N(0,t_j-t_{j-1})$ along every finite strictly increasing list.

Let $A$ be the single probability-one event on which $t\mapsto B_t(\omega )$ is continuous. For $\omega \in A$, the map $t\mapsto a{B}_{ct}(\omega )$ is continuous as a composition of continuous scalar and time maps. Thus the same event $A$ is a common continuity event for $Y$. Together with steps 1.1 and 2.1, this proves that $Y$ is standard Brownian motion.

Define ${\widehat{Y}}_t=Y_t$ on $A$ and ${\widehat{Y}}_t=0$ on $A^c$. Every path of $\widehat{Y}$ is continuous. For each fixed $t$, the random variable ${\widehat{Y}}_t$ is measurable because $A$ is measurable and $Y_t$ is measurable. Since the Borel sigma-algebra of continuous path space is generated by the coordinates, [F3] makes $\omega \mapsto (t\mapsto {\widehat{Y}}_t(\omega ))$ a Borel random element, and its law is a probability. Its coordinate process has the same finite-dimensional laws as $Y$, because the two processes agree on $A$; hence it is centered Gaussian with covariance $\min (s,t)$ by steps 1.1 and 2.1. Uniqueness in [F3] identifies that probability with Wiener measure. When $c=1$, $a=1$ and the process is unchanged; $c=0$ is excluded because neither $c^{-1\mathrm{/}2}$ nor the claimed covariance normalization is defined. AC is used only through [F1] and [F3].