Full and observed data under a missingness pattern

Let $(\mathrm{\Omega },\mathcal{F},P)$ be a probability space and let $W=(W_1,\dots ,W_k)$, for a finite $k\ge 1$, be a random element of the product measurable space $\mathcal{W}={\prod }_{j=1}^k{\mathcal{W}}_j$ (Random elements and real random variables, Measurable spaces and measurable sets). The law of $W$ is the pushforward probability measure defined in Law or distribution of a random element. Let $\mathcal{R}$ be a nonempty finite set of observation patterns with its discrete sigma-algebra, and let $R:\mathrm{\Omega }\to \mathcal{R}$ be a random element.

For each $r\in \mathcal{R}$, specify the recorded coordinate set $I_r\subseteq \{1,\dots ,k\}$ and its complement $J_r=\{1,\dots ,k\}\setminus I_r$. The observed projection $W_{\mathrm{o}\mathrm{b}\mathrm{s}}(r)$ is the coordinate projection into ${\mathcal{W}}_{\mathrm{o}\mathrm{b}\mathrm{s}}(r):={\prod }_{j\in I_r}{\mathcal{W}}_j$, and the missing projection $W_{\mathrm{m}\mathrm{i}\mathrm{s}}(r)$ is the projection into ${\mathcal{W}}_{\mathrm{m}\mathrm{i}\mathrm{s}}(r):={\prod }_{j\in J_r}{\mathcal{W}}_j$. An empty coordinate tuple takes values in a one-point measurable space. The observed record is $O=(R,W_{\mathrm{o}\mathrm{b}\mathrm{s}}(R))$ in the disjoint union ${⨆}_{r\in \mathcal{R}}(\{r\}\times {\mathcal{W}}_{\mathrm{o}\mathrm{b}\mathrm{s}}(r))$, where a set in the disjoint-union sigma-algebra is measurable exactly when each pattern section is measurable. The pattern records which coordinates are available; it does not replace an unrecorded full-data value by zero.