Conditional characteristic-function criterion for Brownian increments ($\mathbb{E}[e^{i\theta (X_t-X_s)}\mid {\mathcal{F}}_s]=e^{-{\theta }^2(t-s)/2}$)

For an adapted continuous process starting at zero, the displayed conditional characteristic function identifies every increment as $N(0,t-s)$ and makes it independent of the preceding sigma-field. Multiply by the indicator of an event in that sigma-field, then use the uniqueness theorem for characteristic functions for finite measures. Thus the process is a Brownian motion in the given filtration.