Consider the completion $(\mathbb{R}^{[0,1]}, \mathcal{B}, \mu)$ of the Wiener measure on $\mathbb{R}^{[0,1]}$ (with the cylinder set $\sigma$-algebra).

Is the following true :

- $C([0,1])\in \mathcal{B}$ ?

I am aware that $\mu^*(C([0,1]))=1$ where $\mu^*$ is the outer measure associated to the Wiener measure, but since the outer measure is not additive, we can't conclude that $C([0,1])$ is measurable for the outer measure. Is it true nonetheless ?