Characterizations of Brownian motion

Question

Let $(\Omega,\mathcal F, P)$ be a probability space, and $\mathcal (L^2,\mathcal G)$ the space of all square integrable functions from $T:=[0,\infty)$ to $\mathbb R$ equipped with the smallest $\sigma$-algebra $\mathcal G$ that makes the coordinate projections measurable. Let $B$ be a Brownian motion (i.e. a function $\Omega\rightarrow L^2$ that is $\mathcal F$-$\mathcal G$-measurable, and the pushforward measure $W:= B_\#P$ (called Wiener measure) satisfies certain conditions).

There are many equivalent characterizations of $B$. According to Wikipedia, there are:

• Characterization by independent and stationary increments:

  1. $B_0 = 0$ almost surely,
  2. $B_t$ is almost surely continuous for every $0\leq t$,
  3. the increments $B_{t} - B_{s}$ are $\mathcal N(0, t-s)$-distributed for every $0\leq s\leq t$, and
  4. the increments $B_{t_1} - B_{t_0}, B_{t_2} - B_{t_1}, \dots, B_{t_n} - B_{t_{n-1}}$ with $t_0<t_1<t_2<\dots<t_{n-1}<t_n$ are independent

• Characterization as Gaussian process:

  1. B is a Gaussian process,
  2. $B_t$ is almost surely continuous for every $0\leq t$,
  3. $E[B_t] = 0$, and
  4. $E[B_tB_s] = s$ for every $0\leq s\leq t$.

• Karhunen-Loève characterization: Let $\xi_1,\xi_2,\dots$ be independent and $\mathcal N(0,1)$-distributed random variables. Then $$B_t = \sum_{k=1}^\infty\sqrt 2\xi_k\frac{\sin((k- 0.5)\pi t)}{(k-0.5)\pi}$$ for every $0\leq t\leq 1$.

I can prove that the first and second characterization are equivalent. For the third characterization I do not know how to adjust for the fact that the representation is only valid in the interval $[0,1]$. How can I extend this to the full time domain $T$?