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# 数学代写|凸分析和最优控制代写Convex Analysis and Optimal Control代考|MATH4071 New fundamentals of Young measure convergence

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## 数学代写|凸分析和最优控制代写Convex Analysis and Optimal Control代考|New fundamentals of Young measure convergence

This paper presents a new, penetrating approach to Young measure convergence in an abstract, measure theoretical setting. It was started in $[12,13,14]$ and given its definitive shape in $[18,22]$. This approach is based on $K$-convergence, a device by which narrow convergence on $\mathcal{P}\left(\mathbf{R}^d\right)$ can be systematically transferred to Young measure convergence. Here $\mathcal{P}\left(\mathbf{R}^d\right)$ stands for the set of all probability measures on $\mathbf{R}^d$ (in the sequel, a much more general topological space $S$ is used instead of $\mathbf{R}^d$ ). Recall that in this context Young measures are measurable functions from an underlying finite measure space $(\Omega, \mathcal{A}, \mu)$ into $\mathcal{P}\left(\mathbf{R}^d\right)$. Recall also from [12], [13] (see also [24]) that $K$-convergence takes the following form when applied to Young measures (see Definition 3.1): A sequence $\left(\delta_k\right)$ of Young measures $K$-converges to a Young measure $\delta_0$ [notation: $\delta_k \stackrel{K}{\longrightarrow} \delta_0$ ] if for every subsequence $\left(\delta_{k_j}\right)$ of $\left(\delta_k\right)$ the following pointwise Cesaro-type convergence takes places
$$\frac{1}{N} \sum_{j=1}^N \delta_{k_j}(\omega) \Rightarrow \delta_0(\omega) \text { as } N \rightarrow \infty$$
at $\mu$-almost every point $\omega$ in $\Omega$. Here ” $\Rightarrow$ ” means classical narrow convergence on $\mathcal{P}\left(\mathbf{R}^d\right)$ (see Definition 2.1). As is shown much more completely in Proposition $3.6$ and Theorem $4.8$, the following fundamental relationship holds between Young measure convergence, denoted by ” $\Longrightarrow$ “, and $K$-convergence as just defined [18, Corollary $3.16]$ :
Theorem $1.1$ Let $\left(\delta_n\right)$ be a sequence of Young measures. The following are equivalent:
(a) $\delta_n \Longrightarrow \delta_0$
(b) Every subsequence $\left(\delta_{n^{\prime}}\right)$ of $\left(\delta_n\right)$ contains a further subsequence $\left(\delta_{n^{\prime \prime}}\right)$ such that $\delta_{n^{\prime \prime}} \stackrel{K}{\longrightarrow} \delta_0$

## 数学代写|凸分析和最优控制代写Convex Analysis and Optimal Control代考|Narrow convergence of probability measures

This section recapitulates some results on narrow convergence of probability measures on a metric space; cf. $[2,27,28,35,46]$. Let $S$ be a completely regular Suslin space, whose topology is denoted by $\tau$. On such a space there exists a metric $\rho$ whose topology $\tau_\rho$ is not stronger than $\tau$, with the property that the Borel $\sigma$-algebras $\mathcal{B}\left(S, \tau_\rho\right)$ and $\mathcal{B}(S, \tau)$ coincide. To see this, recall that in a completely regular space the points are separated by the collection $\mathcal{C}b(S, \tau)$ of all bounded continuous functions on $S$. Since $S$ is also Suslin, it follows by [32, III.32] that there exists a countable subset $\left(c_i\right)$ of $\mathcal{C}_b(S, \tau)$, with $\sup {x \in S}\left|c_i(x)\right|=1$ for each $i$, that still separates the points of $S$. A metric as desired is then given by $\rho(x, y):=\sum_{i=1}^{\infty} 2^{-i}\left|c_i(x)-c_i(y)\right|$. This is because $\tau_\rho \subset \tau$ is obvious and by another well-known property of Suslin spaces, the Borel $\sigma$-algebras $\mathcal{B}(S, \rho)$ and $\mathcal{B}(S, \tau)$ coincide [51, Corollary 2, p. 101]. Of course, if $S$ is a metrizable Suslin space to begin with, then for $\rho$ one can simply take any metric on $S$ that is compatible with $\tau$.
As a consequence of the above, we shall write from now on
$$\mathcal{B}(S):=\mathcal{B}(S, \rho)=\mathcal{B}(S, \tau), \mathcal{P}(S):=\mathcal{P}(S, \rho)=\mathcal{P}(S, \tau)$$
for respectively the Borel $\sigma$-algebra and the set of all probability measures on $(S, \mathcal{B}(S))$.

## 数学代写|凸分析和最优控制代写Convex Analysis and Optimal Control代考|New fundamentals of Young measure convergence

$$\frac{1}{N} \sum_{j=1}^N \delta_{k j}(\omega) \Rightarrow \delta_0(\omega) \text { as } N \rightarrow \infty$$
Young 测度收敛之间存在以下基本关系，记为 ” $\Longrightarrow “$ ， 和 $K$-刚刚定义的收敛[18，推论 $3.16]:$

(a) $\delta_n \Longrightarrow \delta_0$
(b) 每个子序列 $\left(\delta_{n^{\prime}}\right)$ 的 $\left(\delta_n\right)$ 包含另一个子序列 $\left(\delta_{n^{\prime \prime}}\right)$ 这样 $\delta_{n^{\prime \prime}} \stackrel{K}{\longrightarrow} \delta_0$

## 数学代写|凸分析和最优控制代写Convex Analysis and Optimal Control代 考|Narrow convergence of probability measures

$$\mathcal{B}(S):=\mathcal{B}(S, \rho)=\mathcal{B}(S, \tau), \mathcal{P}(S):=\mathcal{P}(S, \rho)=\mathcal{P}(S, \tau)$$

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