Progressively Measurable Process

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $T>0$ be a real number, and let $(\mathcal{F}_t)_{t\in[0,T]}$ be a filtration on $(\Omega,\mathcal{F},P)$ with time index restricted to $[0,T]$. For $t\in(0,T]$ let $\mathcal{B}_{[0,t]}$ be the trace Borel $\sigma$-algebra on $[0,t]$, the family of all sets $S\cap[0,t]$ with $S$ a member of the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$; and let $\mathcal{B}_{[0,0]}$ be the $\sigma$-algebra $\{\emptyset,\{0\}\}$ on $[0,0]=\{0\}$.

A family $X=(X_t)_{t\in[0,T]}$ of real-valued functions on $\Omega$, regarded as the single function $(s,\omega)\mapsto X_s(\omega)$ on the Cartesian product $[0,T]\times\Omega$, is progressively measurable with respect to $(\mathcal{F}_t)_{t\in[0,T]}$ if for every $t\in[0,T]$ the restriction of this function to $[0,t]\times\Omega$ is measurable with respect to the product $\sigma$-algebra $\mathcal{B}_{[0,t]}\otimes\mathcal{F}_t$ and $\mathcal{B}(\mathbb{R})$.

A family of functions with values in Euclidean space $\mathbb{R}^d$ is progressively measurable when each of its $d$ real-valued component families is.