Borel $\sigma$-algebras on the Skorohod space $D[0,1]$

You show that the ball-sigma-algebra of the uniform topology coincides with the Skorohod-Borel-sigma-algebra. However, the Borel-sigma-algebra of the uniform topology is strictly larger (that can happen because the uniform topology is not separable here). See chapt. 15 of Billingsley's book for more on this.

the Borel sigma algebra (wrt the uniform norm) is not generated by balls. since the uniform norm turns the Banach space into a non-separable one. (i.e. you can not write every Open set U as countable union of balls!)

Regards,

Chiu