fa.functional analysis - Borel(X) = sigma(X') for X non-separable

If $I$ is uncountable, then in space $l^2(I)$ no countable set of functionals separates points. Consequently, for any set $A$ in the sigma-algebra generated by these functionals [the Baire sets for the weak topology, see reference below], if $0 in A$, then an entire subspace is contained in $A$. So all elements of this sigma-algebra are unbounded. Thus this sigma-algebra is not all of the norm-Borel sets.

My papers on measurability in Banach space:
Indiana Univ. Math. J. 26 (1977) 663--677
Indiana Univ. Math. J. 28 (1979) 559--579

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For gaussian measures in Banach space, you really want the example
of Fremlin and Talagrand, "A Gaussian measure on $l^{infty}$". Ann. Probab. 8 (1980), no. 6, 1192--1193. This gaussian measure on $l^infty$ with the cylindrical sigma-algebra has total mass 1, yet every ball of radius 1 has measure 0.