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# 经济代写|计量经济学代写Introduction to Econometrics代考|ECON-322 SELECT IONS AND MEASURAB IL ITY

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## 经济代写|计量经济学代写Introduction to Econometrics代考|SELECT IONS AND MEASURAB IL ITY

It has been common to think of random sets as bundles of random variables – the selections of the random sets. The formal definition follows, where an $\mathfrak{X}$-valued random element $\boldsymbol{x}$ is a measurable map $\boldsymbol{x}: \Omega \mapsto \mathfrak{X}$ where the measurability is understood with respect to the conventional Borel $\sigma$-algebra $\mathcal{B}(\mathfrak{A})$ on the space $\mathfrak{X}$.

Definition 2.1 For any random set $\boldsymbol{X}$, a (measurable) selection of $\boldsymbol{X}$ is a random element $\boldsymbol{x}$ with values in $\mathfrak{X}$ such that $\boldsymbol{x}(\omega) \in \boldsymbol{X}(\omega)$ almost surely. We denote by $\operatorname{Sel}(\boldsymbol{X})$ (also denoted by $\mathbf{L}^0(\boldsymbol{X})$ or $\mathbf{L}^0(\boldsymbol{X}, \mathfrak{R})$ ) the set of all selections from $X$.

We often call $\boldsymbol{x}$ a measurable selection in order to emphasize the fact that $\boldsymbol{x}$ is measurable itself; the notation $\mathbf{L}^0(\boldsymbol{X}, \mathfrak{I})$ emphasizes the fact that selections are measurable with respect to the $\sigma$-algebra $\mathfrak{A}$. Recall that a random closed set is defined on the probability space $(\Omega, \mathfrak{I}, \mathbf{P})$ and, unless stated otherwise, “almost surely” means “P-almost surely.” We often abbreviate “almost surely” as “a.s.” A possibly empty random set clearly does not have a selection, so unless stated otherwise we assume that all random sets are almost surely nonempty. One can view selections as curves evolving in the “tube” being the graph of the random set $\boldsymbol{X}$ (see Figure 2.1).

## 经济代写|计量经济学代写Introduction to Econometrics代考|Existence of Measurable Selections

The Fundamental Selection theorem establishes the existence of a selection for non-empty random closed sets in rather general spaces. It is formulated below for random closed sets in $\mathbb{R}^d$.

Theorem $2.8$ (Fundamental Selection theorem) If $\boldsymbol{X}: \Omega \mapsto \mathcal{F}$ is an almost surely non-empty random closed set in $\mathbb{R}^d$, then $\mathrm{X}$ has a measurable selection.
Proof. Let $\mathbb{Q}=\left{x_i, i \geq 1\right}$ be an enumeration of the set of points with rational coordinates. Define $\boldsymbol{X}0(\omega)=\boldsymbol{X}(\omega)$ and $k_0=1$, and then inductively, if $\boldsymbol{X}_i(\omega)$ is defined for $i=0, \ldots, n$, then let $$k{n+1}(\omega)=\min \left{i \geq 1: \boldsymbol{X}n(\omega) \cap B{(n+1)^{-1}}\left(x_i\right) \neq \emptyset\right},$$
and $\boldsymbol{X}{n+1}(\omega)=\boldsymbol{X}_n(\omega) \cap B{(n+1)^{-1}}\left(x_{k_{n+1}}\right)$. Then $\left{\boldsymbol{X}_n(\omega), n \geq 0\right}$ is a nonincreasing sequence of closed sets such that the diameter of $X_n$ is at most $2 / n$. By the completeness of $\mathbb{R}^d$, the sequence $\left{\boldsymbol{X}_n, n \geq 0\right}$ has a non-empty intersection, which is then necessarily a singleton denoted by ${\boldsymbol{x}(\omega)}$.

Note that $\boldsymbol{X}0$ is a random closed set and assume that $\boldsymbol{X}_1, \ldots, \boldsymbol{X}_n$ are random closed sets too. For each $m \geq 1$, $$\left{k{n+1}=m\right}=\bigcap_{i=1}^{m-1}\left{\boldsymbol{X}n \cap B{(n+1)^{-1}}\left(x_i\right)=\emptyset\right} \cap\left{\boldsymbol{X}n \cap B{(n+1)^{-1}}\left(x_m\right) \neq \emptyset\right} \in \mathfrak{A} .$$
Thus, for each closed $F$,
$$\boldsymbol{X}{n+1}^{-}(F)=\bigcup{m \geq 1} \boldsymbol{X}n^{-}\left(B{(n+1)^{-1}}\left(x_m\right)\right) \cap\left{k_{n+1}=m\right} \in \mathfrak{A},$$
where $\boldsymbol{X}^{-}$is defined in (1.1), so that $\boldsymbol{X}{n+1}$ is measurable by Theorem $2.10$ below. By induction, $\boldsymbol{X}_n(\omega)$ is a random closed set for all $n$. Since $${x}=\bigcap{n \geq 0} X_n,$$
the random singleton ${x}$ is measurable by Theorem $2.10$, which is equivalent to the measurability of $\boldsymbol{x}$.

## 经济代写|计量经济学代写Introduction to Econometrics代考|Existence of Measurable Selections

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$$x=\bigcap n \geq 0 X_n,$$

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