I'm going over some materials on stochastic analysis, and stuck with a problem on Gaussian white noise:
Let $(\mathbb{R}^d,\mathcal{B},m)$ be the Borel measurable space on $\mathbb{R}^d$. A white noise is a mean zero Gaussian process $\{W(A):A\in\mathcal{B},m(A)<\infty\}$ with covariance $\mathbb{E}[W(A)W(B)]=m(A\cap B)$.
Fix positive numbers $\{c_k\}_{k\in\mathbb{N}}$ such that $\sum c_k^2=1$ but $\sum c_k=\infty$. Let $A=\bigcup_k A_k$ be a disjoint union of Borel sets with $m(A_k)=c_k^2$. Show that $\sum|W(A_k)|=\infty$ a.s.
Even though one can show that $\sum W(A_k)=W(A)$ a.s. by a standard result on random series, the problem above shows that $\sum W(A_k)$ doesn't converge absolutely and hence $W(\cdot,\omega)$ cannot be a signed measure.
Any thought will be appreciated.