# 数学代写|凸分析和最优控制代写Convex Analysis and Optimal Control代考|MATH3971 K-convergence of Young measures

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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|K-convergence of Young measures

A Young measure is a function $\delta: \Omega \rightarrow \mathcal{P}(S)$ that is measurable with respect to $\mathcal{A}$ and $\mathcal{B}(\mathcal{P}(S))$ The set of all such Young measures is denoted by $\mathcal{R}(\Omega ; S)$. By $\mathcal{B}(S)=$ $\mathcal{B}\left(S, \tau_\rho\right)$ of the previous section it is not hard to see that Young measures are precisely the transition probabilities from $(\Omega, \mathcal{A})$ into $(S, \mathcal{B}(S))$ [45, III.2], i.e., $\delta: \Omega \rightarrow \mathcal{P}(S)$ belongs to $\mathcal{R}(\Omega ; S)$ if and only if $\omega \mapsto \delta(\omega)(B)$ is $\mathcal{A}$-measurable for every $B \in \mathcal{B}(S)$.

For some elementary measure-theoretical properties of Young measures the reader is referred to [45, III.2] or $[2,2.6]$. In particular, the product measure induced on $(\Omega \times S, \mathcal{A} \times \mathcal{B}(S))$ by $\mu$ and any $\delta \in \mathcal{R}(\Omega ; S)$ (cf. [45, III.2]) is denoted by $\mu \otimes \delta$. Let $\mathcal{L}^0(\Omega ; S)$ be the set of all measurable functions from $(\Omega, \mathcal{A})$ into $(S, \mathcal{B}(S))$. A Young measure $\delta \in \mathcal{R}(\Omega ; S)$ is said to be Dirac if it is a degenerate transition probability [4.5, III.2], i.e., if there exists a function $f \in \mathcal{L}^0(\Omega ; S)$ such that for every $\omega$ in $\Omega$
$$\delta(\omega)=\epsilon_{f(\omega)}:=\text { Dirac measure at the point } f(\omega) .$$
Conversely, $\delta$ is also called the Young measure relaxation of $f$. In this special case $\delta$ is denoted by $\delta=\epsilon_f$. The set of all Dirac Young measures in $\mathcal{R}(\Omega ; S)$ is denoted by $\mathcal{R}_{\text {Dirac }}(\Omega ; S)$

The fundamental idea behind Young measure theory is that, in some sense, $\mathcal{R}(\Omega ; S)$ forms a completion of $\mathcal{L}^0(\Omega ; S)$, when the latter is identified with $\mathcal{R}_{\text {Dirac }}(\Omega ; S)$.

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

In this section our program to transfer narrow convergence results for probability measures (section 2) to Young measures comes is completed. We use the same fundamental hypotheses as in the previous section: $(\Omega, \mathcal{A}, \mu)$ is a finite measure space and $(S, \tau)$ is a completely regular Suslin space, on which we also consider the weak metric topology $\tau_\rho \subset \tau$. We start out by giving the definition of narrow convergence for Young measures [3, 4, 10] (see also [38]).

Definition $4.1$ (narrow convergence in $\mathcal{R}(T ; S)$ ) A sequence $\left(\delta_n\right)$ in $\mathcal{R}(\Omega ; S)$ converges $\tau$-narrowly to $\delta_0$ in $\mathcal{R}(\Omega ; S)$ (this is denoted by $\delta_n \stackrel{\tau}{\Longrightarrow} \delta_0$ ) if for every $A \in \mathcal{A}$ and $c$ in $\mathcal{C}b(S, \tau)$ $$\lim _n \int_A\left[\int_S c(x) \delta_n(\omega)(d x)\right] \mu(d \omega)=\int_A\left[\int_S c(x) \delta_0(\omega)(d x)\right] \mu(d \omega) .$$ The weaker notion of $\tau\rho$-narrow convergence is defined by replacing $\tau$ by $\tau_\rho$; this is denoted by $” \stackrel{\rho}{\Longrightarrow}$. We shall occasionally use ” $\Longrightarrow “$ in situations where we need not distinguish between the two at all. We shall see that on $\tau$-tight sets of Young measures these two modes actually coincide (note the complete analogy to section 2). For further benefit, note carefully the distinct notation used for narrow convergence for probability measures (indicated by short arrows) and Young measure convergence (indicated by long arrows).

## 数学代写|凸分析和最优控制代写CONVEX ANALYSIS AND OPTIMAL CONTROL代考|K-CONVERGENCE OF YOUNG MEASURES

Young 度量是一个函数 $\delta: \Omega \rightarrow \mathcal{P}(S)$ 这是可以衡量的 $\mathcal{A}$ 和 $\mathcal{B}(\mathcal{P}(S))$ 所有此类 Young 度量的集合表示为 $\mathcal{R}(\Omega ; S)$. 经过 $\mathcal{B}(S)=\mathcal{B}\left(S, \tau_\rho\right)$ 从上一节不难看出，Young 测 度正是从 $(\Omega, \mathcal{A})$ 进入 $(S, \mathcal{B}(S))$
45, III $.2$

$4.5, I I I .2$
, 即如果存在函数 $f \in \mathcal{L}^0(\Omega ; S)$ 这样对于每个 $\omega$ 在 $\Omega$
$\delta(\omega)=\epsilon_{f(\omega)}:=$ Dirac measure at the point $f(\omega)$.

## 数学代写凸分析和最优控制代写CONVEX ANALYSIS AND OPTIMAL CONTROL代考|NARROW CONVERGENCE OF YOUNG MEASURES

$3,4,10$
seealso[38].
$\lim _n \int_A\left[\int_S c(x) \delta_n(\omega)(d x)\right] \mu(d \omega)=\int_A\left[\int_S c(x) \delta_0(\omega)(d x)\right] \mu(d \omega)$.Theweakernotionof $\backslash$ tau \rho-narrowconvergenceisdefinedbyreplacing $\backslash$ 能够by
insituationswhereweneednotdistinguishbetweenthetwoatall. Weshallseethaton \taus-tight 集的 Young 度量这两种模式实际上是重合的
notethecompleteanalogytosection 2 . 为了进一步的好处，请仔细注意用于概率测量的窤收敛的不同符昊indicatedbyshortarrows和 Young 度量收敛 indicatedbylongarrows.

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