In Gardiner's chapter 4.1 he introduces the white noise $\xi(t)$ with properties: $\langle \xi(t) \rangle=0$, $\xi(t)$ is independent of $\xi(t')$ for $t\neq t'$. He says that the last property implies $\langle \xi(t)\xi(t')\rangle=\delta(t-t')$. I get that it should only be different from zero for $t'=t$, but why must it be the delta?
Then he defines the temporal integral of the white noise $u(t)=\int_0^t \xi(t')dt'$. He claims that assuming that $u(t)$ is continuous, then $u(t)$ is a Markov Process by the following argument:
$u(t')=\int_0^t ds \xi(s)+\int_t^{t'} ds \xi(s)=\lim_{\varepsilon \to 0}\int_0^{t-\varepsilon} ds \xi(s)+\int_t^{t'} ds \xi(s)$
For any $\varepsilon>0$, the white noise $\xi(s)$ in the first integral is independent of the white noise $\xi(s)$ in the second. And then he says: by continuity, $u(t)$ (the first integral in the equation above) and $u(t')-u(t)$ (the second integral) are statistically independent. How can we prove this statement? I understand the intuitive argument, but am not sure how to prove it.