# Graph measurability (difficult but interesting)

Let $\mu(\cdot)$ be a probability measure on $W \subseteq \mathbb{R}^p$, so that $\int_W \mu(dw) = 1$.

Consider a locally bounded function $f: \mathbb{R}^n \times \mathbb{R}^m \times W \rightarrow \mathbb{R}^n$ such that $\forall w \in W$ $\ (x,y) \mapsto f(x,y,w)$ is continuous, $\forall (x,y) \in \mathbb{R}^{n} \times \mathbb{R}^m$ $\ w \mapsto f(x,y,w)$ is measurable.

Also consider a locally bounded function $g: \mathbb{R}^n \rightarrow \mathbb{R}^m$.

For $\epsilon: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ continuous and positive definite, define the set-valued maps

$$F(x,y,w) := f( x + \epsilon(x) \bar{\mathbb{B}},y,w) + \epsilon(x) \bar{\mathbb{B}}$$

$$G(x) := g( x + \epsilon(x)\bar{\mathbb{B}} ) + \epsilon(x)\bar{\mathbb{B}}$$

$$H(x,u,w):= \left( \begin{array}{c} F(x,y,w) \\ G( F(x,y,w) ) \end{array} \right)$$

Prove that the map

$$w \mapsto graph\left( H(\cdot,\cdot,w) \right):= \{ (x,y,z) \in \mathbb{R}^n \times \mathbb{R}^m \times \mathbb{R}^{n+m} \mid z \in H(x,y,w) \}$$

is measurable.

Definition of measurability for a set-valued map:

A set-valued mapping $S: T \rightrightarrows \mathbb{R}^q$ is measurable if for every open set $\mathcal{O} \subset \mathbb{R}^q$ the set $S^{-1}(\mathcal{O}) \subset T$ is measurable, i.e. $S^{-1}(\mathcal{O}) \in \mathcal{A}$ (some $\sigma$-field of subsets of $T$). In particular, the set $dom S = S^{-1}(\mathbb{R}^q)$ must be measurable.

-