Definition

Let $X$ be a right continuous $L^{2}$-martingale. For all $0\le s<t, F\in\mathcal{F}_{s}$ we define $$\mu_{X}((s,t]\times F)=E[\mathbb{1}_{F}(X_{t}^{2}-X_{s}^{2})]$$and $$\mu_{X}(\{ 0 \}\times F)=0, \forall F\in\mathcal{F}_{0}$$ This $\mu_{X}$ is called the Doléans measure associated with $X$.

Proposition ($\mu_{X}$ extends to σ-finite $\mathcal{R}$)

The Doléans measure, $\mu_{X}$ extends uniquely to a σ-finite measure $\mathcal{R}$ on $\mathbb{R}^{+}\times\Omega$ generated by sets of the form - $\{ 0 \}\times F_{0}$ with $F_{0}\in\mathcal{F}_{0}$ and; - $(s,t]\times F,0\le s<t, \forall F\in\mathcal{F}_{s}$