Proof of Brownian Motion is an Ito Integrator with Unit Intensity

Step 1 (Square-integrability). By clause (i) of Standard Brownian Motion, $B_0=0$ almost surely, so $P(B_0=0)=1$ and the null-equivalence clause of Square-Integrable Random Variables and the Mean-Square Inner Product gives $\lVert B_0-0\rVert_2=0$; in particular $B_0$ is square-integrable with $\mathbb{E}[B_0]=0$ and $\mathbb{E}[B_0^{2}]=0$. For $t>0$, the increment $B_t-B_0$ is a Gaussian random variable by clause (iii) of Standard Brownian Motion, hence square-integrable by Square-Integrability, Moments, and Covariance Matrix of a Gaussian Random Vector, and $B_t=B_0+(B_t-B_0)$ is square-integrable by the closure properties of Square-Integrable Random Variables and the Mean-Square Inner Product.

Step 2 (Adaptedness and increment independence). $B$ is adapted to its natural filtration, as recorded in Filtration, Adapted Process, and Natural Filtration. By clause (iv) of Standard Brownian Motion, $B$ has independent increments, and $B_0=0$ almost surely; so Increments Are Independent of the Natural Filtration Past (with the constant $c=0$) shows that for all $0\le s<t$ the $\sigma$-algebras $\sigma(B_t-B_s)$ and $\mathcal{F}^{B}_s$ are independent. This is clause (iii) of Ito Integrator of Intensity Type for the filtration $(\mathcal{F}^B_t)_{t\ge0}$.

Step 3 (Martingale property). Let $0\le s\le t$ and $A\in\mathcal{F}^{B}_s$; we may assume $s<t$. The indicator $\mathbf{1}_{A}$ is an $\mathcal{F}^{B}_s$-measurable random variable (its preimages are among $\emptyset$, $A$, $\Omega\setminus A$, $\Omega$), so by the consequence clause of Increments Are Independent of the Natural Filtration Past, $B_t-B_s$ and $\mathbf{1}_A$ are independent. Both are integrable (square-integrable random variables are integrable by Square-Integrable Random Variables and the Mean-Square Inner Product; $\mathbf{1}_A$ is bounded), so Expectation of a Product of Independent Random Variables and the mean-zero clause (iii) of Standard Brownian Motion give

hence $\mathbb{E}[B_t\mathbf{1}_A]=\mathbb{E}[B_s\mathbf{1}_A]$ by linearity of expectation (Linearity and Monotonicity of the Lebesgue Integral). Together with Steps 1 and 2, this is the averaged form of the martingale property, so $B$ is a square-integrable martingale with respect to $(\mathcal{F}^{B}_t)_{t\ge0}$ by the equivalence recorded there.

Step 4 (Intensity). The constant function $\rho\equiv1$ (extended by $0$ off $[0,\infty)$ as in Ito Integrator of Intensity Type) is measurable, and $\int_{\mathbb{R}}\mathbf{1}_{(0,T]}\rho\,d\lambda=\lambda((0,T])=T<\infty$ for every $T>0$, since the integral of an indicator equals the measure of the set (Simple Function and Its Integral) and Lebesgue measure assigns to an interval its length. For $0\le s<t$, clause (iii) of Standard Brownian Motion gives $\mathbb{E}[B_t-B_s]=0$ and $\operatorname{Var}(B_t-B_s)=t-s$, so with the variance identity,

This is clause (iv) of Ito Integrator of Intensity Type.

Clauses (i)-(iv) of Ito Integrator of Intensity Type are verified (clause (ii) is Step 1), so $(B,1)$ is an It^{o} integrator of intensity type with respect to $(\mathcal{F}^{B}_t)_{t\ge0}$. $\square$