Conditional Expectations: Notation

(i) Base. This setting extends Probability Spaces: Notation: its conventions and notation are in force. Unless stated otherwise, $\mathcal{G}$ denotes a $\sigma$-algebra on $\Omega$ with $\mathcal{G}\subseteq\mathcal{F}$.

(ii) $\mathcal{G}$-measurable and Borel functions. Let $n,p,q\in\mathbb{N}$. For a $\sigma$-algebra $\mathcal{G}$ on $\Omega$ with $\mathcal{G}\subseteq\mathcal{F}$, a function from $\Omega$ into $\mathbb{R}$, $\mathbb{R}^{n}$ or $\mathbb{R}^{p\times q}$ is $\mathcal{G}$-measurable if it is $\mathcal{G}/\mathcal{B}(\mathbb{R})$-, $\mathcal{G}/\mathcal{B}(\mathbb{R}^{n})$-, respectively $\mathcal{G}/\mathcal{B}(\mathbb{R}^{p\times q})$-measurable, as in Measurable Map, and the Set of Measurable Maps (iii), Measurable Map, and the Set of Measurable Maps (iv) and Measurable Map, and the Set of Measurable Maps (v) on $(\Omega,\mathcal{G})$. For a metric space $(S,d)$, a function from $S$ into $\mathbb{R}$ or $\mathbb{R}^{n}$ is Borel if it is $\mathcal{B}(S)/\mathcal{B}(\mathbb{R})$-, respectively $\mathcal{B}(S)/\mathcal{B}(\mathbb{R}^{n})$-measurable, as in Measurable Map, and the Set of Measurable Maps (iii) and Measurable Map, and the Set of Measurable Maps (iv) on $(S,\mathcal{B}(S))$.

(iii) Products. For $n\in\mathbb{N}$, functions $W:\Omega\to\mathbb{R}$ and $U,V:\Omega\to\mathbb{R}^{n}$, and $R\in\mathbb{R}^{n\times n}$, $WU$, $U\cdot V$ and $U\cdot(RV)$ are the functions

$$\omega\mapsto W(\omega)U(\omega),\qquad\omega\mapsto U(\omega)\cdot V(\omega),\qquad\omega\mapsto U(\omega)\cdot\bigl(RV(\omega)\bigr)$$

on $\Omega$, formed with the dot product and the matrix-vector product (Maps Formed Pointwise from Vector and Matrix Operations (i), Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii)).