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# 数学代写|随机分析代写Stochastic Calculus代考|MATH581 Orthogonally scattered measures and dilation

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## 数学代写|随机分析代写Stochastic Calculus代考|Orthogonally scattered measures and dilation

In the previous subsection we saw that $\nu$-boundedness is equivalent to the existence of a weak Radon-Nikodým derivative for an $H$-valued measure. In this subsection, we take another view for the existence.

Let $\mathcal{H}$ be any Hilbert space (not necessarily separable) and consider $\mathcal{H}$-valued measures. A special class of Hilbert space measures is that of orthogonally scattered measures. An extensive study of such measures is done by Masani [21]. An important theorem is that any Hilbert space valued measure has an orthogonally scattered dilation. Using this fact we can give fairly simple sufficient conditions for the existence of the weak RadonNikodým derivative, which is our object in this subsection.

Definition 2.4.1. An $\mathcal{H}$-valued measure $\eta \in c a(\mathfrak{A}, \mathcal{H})$ is said to be orthogonally scattered if $(\eta(A), \eta(B))_{\mathcal{H}}=0$ for every disjoint $A, B \in \mathfrak{A}$. Let caos $(\mathfrak{A}, \mathcal{H})$ denote the set of all $\mathcal{H}$-valued orthogonally scattered measures on $\mathfrak{A}$.
As is easily seen, the measures $\xi$ given in Examples 2.2 .1 and 2.2 .2 are orthogonally scattered.

Now let $\eta \in \operatorname{caos}(\mathfrak{A}, \mathcal{H})$ be an $\mathcal{H}$-valued orthogonally scattered measure. Define $\nu_\eta$ by
$$\nu_\eta(A)=|\eta(A)|_{\mathcal{H}}^2, \quad A \in \mathfrak{A} .$$

## 数学代写|随机分析代写Stochastic Calculus代考|Dunford-Schwartz type integration

We define an integral of a scalar function with respect to a Hilbert spacevalued measure with a weak Radon-Nikodým derivative. Then, we study a relationship between this type of integral and Dunford-Schwartz integral. As before we assume that $H$ is a separable Hilbert space.

Definition 2.5.1. Let $\xi \in c a(\mathfrak{A}, H)$ be an $H$-valued measure such that $\xi \ll \nu$, where $\nu$ is a $\sigma$-finite measure on $\mathfrak{A}$. Assume that $\xi$ has a weak Radon-Nikodým derivative $\xi^{\prime}$ with respect to $\nu$. A scalar valued measurable function $f$ on $\Theta$ is said to be $\xi$-integrable if there exists a sequence $\left{f_n\right}_{n=1}^{\infty} \subset L^0(\Theta)$ of scalar valued $\mathfrak{A}$-simple functions on $\Theta$ such that
(1) $\left|f_n \xi^{\prime}-f \xi^{\prime}\right|_H \rightarrow 0 \nu$-a.e. as $n \rightarrow \infty$;
(2) For every $A \in \mathfrak{A}$ the sequence $\left{\int_A f_n d \xi\right}_{n=1}^{\infty}$ is a Cauchy sequence in $H$, where the integral $\int_A f_n d \xi$ is defined in an obvious way.
In this case we shall write
$$\int_A f d \xi=\int_A f \xi^{\prime} d \nu=\lim _{n \rightarrow \infty} \int_A f_n d \xi, \quad A \in \mathfrak{A},$$
which is called the integral of $f$ with respect to $\xi$ over $A$. Let $L^1(\xi)$ denote the set of all scalar valued functions on $\Theta$ that are $\xi$-integrable.

## 数学代写|随机分析代写Stochastic Calculus代写|Orthogonally scattered measures and dilation

$$\nu_\eta(A)=|\eta(A)|_{\mathcal{H}}^2, \quad A \in \mathfrak{A}$$

## 数学代写|随机分析代写Stochastic Calculus代考|Dunford-Schwartz type integration

(1) $\left|f_n \xi^{\prime}-f \xi^{\prime}\right|H \rightarrow 0 \nu-\mathrm{ae}$ 作为 $n \rightarrow \infty$; (2) 对于每个 $A \in \mathfrak{A}$ 序列 $\backslash$ left 缺少或无法识别的分隔符 是一个柯 西序列 $H$ ，其中积分 $\int_A f_n d \xi$ 以明显的方式定义。 在这种情况下，我们将写 $$\int_A f d \xi=\int_A f \xi^{\prime} d \nu=\lim {n \rightarrow \infty} \int_A f_n d \xi, \quad A \in \mathfrak{A}$$

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