Levy characterization of Brownian motion

The conditional law of the increment is $N(0,t-s)$: by [F1], $E[e^{i\theta (M_t-M_s)}\mid {\mathcal{F}}_s]=e^{-{\theta }^2(t-s)\mathrm{/}2}$ for every real $\theta$; for each $A\in {\mathcal{F}}_s$, [F4] gives $E[1_A{e}^{i\theta (M_t-M_s)}]=E[1_A]e^{-{\theta }^2(t-s)\mathrm{/}2}$. Let $Q_A(\mathrm{\Gamma }):=E[1_A{1}_{\{M_t-M_s\in \mathrm{\Gamma }\}}]$ for Borel $\mathrm{\Gamma }$, a finite measure of total mass $P(A)$; its Fourier transform is $Q_A$'s transform $E[1_A{e}^{i\theta (M_t-M_s)}]=P(A){\varphi }_{t-s}(\theta )$ with ${\varphi }_{t-s}$ the characteristic function of $N(0,t-s)$ by [F2]. Since $Q_A$ and $P(A)N(0,t-s)(\cdot )$ are finite Borel measures with equal Fourier transforms, [F3] applied after normalization (or to the differences) gives $Q_A(\mathrm{\Gamma })=P(A)N(0,t-s)(\mathrm{\Gamma })$ for all Borel $\mathrm{\Gamma }$.

Independence: taking $A=\mathrm{\Omega }$ in step 1.1 gives the marginal law $P(M_t-M_s\in \mathrm{\Gamma })=N(0,t-s)(\mathrm{\Gamma })$. For general $A\in {\mathcal{F}}_s$ and Borel $\mathrm{\Gamma }$, step 1.1 therefore gives $E[1_A{1}_{\{M_t-M_s\in \mathrm{\Gamma }\}}]=P(A)P(M_t-M_s\in \mathrm{\Gamma })$, which is exactly the independence of $M_t-M_s$ from ${\mathcal{F}}_s$, together with the stated law.

Finite lists of increments: for $0=t_0<t_1<\cdots <t_n$ the increments $M_{t_j}-M_{t_{j-1}}$ are independent with laws $N(0,t_j-t_{j-1})$. Induction on $n$: for $n=1$ this is step 2.1; given the claim for $n$ increments, the conditional law of the increment at $t_{n+1}$ given ${\mathcal{F}}_{t_n}$ is $N(0,t_{n+1}-t_n)$ by step 2.1 and is independent of ${\mathcal{F}}_{t_n}$, hence independent of the sigma-algebra generated by the previous increments (which is contained in ${\mathcal{F}}_{t_n}$), and the tower property [F4] multiplies the joint law.

Conclusion and boundary cases: $M$ is adapted, has continuous paths and $M_0=0$ almost surely, and steps 2.1 and 3.1 verify clauses 2 and 3 of the definition of a standard Brownian motion and the increment condition (H) relative to $({\mathcal{F}}_t)$; hence $M$ is a standard Brownian motion with the stated filtration property. At $s=t$ the increment is $0$ with law $N(0,0)$ and independence is trivial; at $s=0$ the identity gives the law of $M_t$; for $\theta =0$ the conditional identity is the trivial constant-$1$ identity; if the clock were $ct$ with $c>0$, rescaling would give the Gaussian factor $e^{-c{\theta }^2(t-s)\mathrm{/}2}$ and the same argument with variance $c(t-s)$; a non-continuous local martingale is not covered, since continuity is used both from the lemma and in the definition of Brownian motion; and AC enters only through [F5].