Levy's Characterization theorem states that if $M$ is a continuous local martingale such that $M_0=0$ and the process $M_t^2-t$ is a continuous local martingale, then $M$ is a Brownian motion.
A standard approach (Karatzas&Shreve, page 157) to prove the above is to show that \begin{equation} \tag{1} E[e^{iu(M_t-M_s)}|\mathcal{F}_s]=e^{-u^2(t-s)/2} \end{equation} and to do so we take $A\in\mathcal{F}_s$ and applying Ito formula to $1_{A}\exp(iux)$ taking expectation and solving an ODE gives us, $$ \tag{2} E[1_{A}e^{iu(M_t-M_s)}]=P(A)e^{-u^2(t-s)/2}. $$
My question is which equation (1) or (2) tells us independence of increments? And how does (2) implies (1)?
(2) implies (1) by definition of conidtional expectation: $E(X|\mathcal G)=c$ (where $c$ non-random) if and only if $E(I_A X)=E(cI_A)\equiv cP(A)$ for all $A \in \mathcal G$.
(2) also implies independence of $\mathcal F_s$ and $M_t-M_s$ and this implies that $(M_t)$ has independent increments.
[This is not so obvious. What you do is take any random variable $Z$ measurable w.r.t. $\mathcal F_s$ and show that the joint characteristic function of $Z$ and $M_t-M_s$ is the product of the individual characteristic functions. This gives independence by two dimensional inversion theorem].