Random measure and independence of a disjoint family

Question

I'm reading about Random Measures.

As a definition, the book I'm reading gives that:

Let $(E, \mathcal{E})$ be a measurable space. A random measure on $(E, \mathcal{E})$ is a transition kernel from $(\Omega, H)$ into $(E, \mathcal{E}).$

More explicitly, a mapping $M:\Omega\times\mathcal{E}\rightarrow\mathbb{R}^{+}$ is called a random measure if $\omega → M(\omega, A)$ is a random variable for each $A \in\mathcal{E}$ and if $A\rightarrow M(\omega,A)$ is a measure on $(E, \mathcal{E})$ for each $\omega\in\Omega.$

Then, comes the following definition:

If $M$ is a random measure on $(E,\mathcal{E}),$ or each $A\in\mathcal{E},$ we define the $\sigma-$algebra $$\mathcal{F}(A)=\sigma(\{M(B):B\in\mathcal{E}, B\subset A\}).$$

And finally the next observation: If $\{A_{i}\}_{i=1}^{n}$ are disjoint sets on $\mathcal{E},$ then $\sigma-$algebras $\mathcal{F}_{A_{1}},\ldots,\mathcal{F}_{A_{n}}$ are independent.

I don't get why the random variables $M(A_{1}),\ldots,M(A_{n})$ are independent.

I've tried to use the definition above but I don't get any useful.

Any kind of help is thanked in advanced.

Answer 1

This is obviously false: take $M(\omega,A)=\mu(A) X(\omega)$ where $X$ is a fixed positive random variable. A random measure which satisfies the independence condition stated above is called an independently scattered random measure.