Definition of random measures

Question

Introducing the notion of a random measure, textbooks usually start with a locally compact second countable Hausdorff space. Where does this requirement come from?

I would like to have a motivation for this requirement. That is, I would like to define a random measure on a general measurable space $(X,\mathcal{B})$ simply as a kernel from a probability space to $(X,\mathcal{B})$. Additional structure of $(X,\mathcal{B})$ should then be motivated by e.g. counterintuitive examples. For example, in the book of Last and Penrose (2017) (link to pdf), Exercise 2.5 yields that point measures usually do not have the representation with Dirac measures.

Are there other examples, (intuitive) motivations and reasons to use a locally compact (!) second countable (!) Hausdorff space?

Once a locally compact second countable Hausdorff space $(X,\mathcal{T})$ is given, the theory of random measures just considers measures $\mu$ that are locally finite, that is, $\mu(K) < \infty$ for all relatively compact sets $K \subset X.$ Therefore, I wonder the following:

Let $\mathcal{B} = \sigma(\mathcal{T})$ be the Borel-$\sigma$-algebra on $X,$ let $M_X$ be the set of all measures on $(X,\mathcal{B})$ and let $\mathcal{M}_X$ be the $\sigma$-algebra generated by the evaluation maps $\mu \mapsto \mu(B)$ for all $B \in \mathcal{B}.$

Is the set of all locally finite measures $\mathcal{M}_X$-measurable?

Answer 1

By way of introduction:

As expressed in some of the comments, I find the "locally compact" assumption possibly a bit too strong.

A weaker assumption than having a locally compact second-countable Hausdorff space would be that the space is a Lusin space, i.e. a separable metrisable topological space that satisfies the following beautifully equivalent statements:

• $X$ can be topologically identified with a Borel subset of a completely metrisable topological space;
• for any metrisation $d$ of the topology of $X$, $X$ is a Borel subset of the $d$-completion of $X$;
• if $X$ is uncountable then $(X,\mathcal{B}(X))$ is measurably isomorphic to $([0,1],\mathcal{B}([0,1]))$.

[I emphasise that this equivalence assumes that $X$ is a separable metrisable space.]

If $X$ is a Lusin space then we have some very nice properties in regards to random measures. But I will also explain some of the important properties more generally of second countable spaces, and of separable metrisable spaces.

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1. Countably generated Borel space

The first thing to say is that everything is pretty much hopeless if you're working on a space $X$ for which the Borel $\sigma$-algebra $\mathcal{B}(X)$ is not countably generated. Much of probability theory is about almost sure statements, and it's often really important to be able to move logically from

"for each $A \in \mathcal{B}(X)$, some assertion is almost surely true"

to

"it's almost surely true that for every $A \in \mathcal{B}(X)$ the assertion holds".

The ability to do this often relies on $\mathcal{B}(X)$ being countably generated. This is fundamental to why second countability (or equivalently, for metrisable spaces, separability) is assumed - it guarantees that the Borel $\sigma$-algebra is countably generated.

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2. Measurable structure on the space of probability measures

It would be really good if, on the space $M_X$ of Borel probability measures on $X$, the evaluation $\sigma$-algebra $\sigma(\mu \mapsto \mu(A) : A \in \mathcal{B}(X))$ is a "nice" $\sigma$-algebra to work with. We have the following:

Theorem 1. Let $X$ be a Lusin space (resp. any separable metrisable space). Then $M_X$ equipped with the topology of weak convergence is also a Lusin space (resp. a separable metrisable space), and the Borel $\sigma$-algebra of the topology of weak convergence is precisely the evaluation $\sigma$-algebra.

An immediate corollary is that if $X$ is a separable metrisable space then the evaluation $\sigma$-algebra is countably generated.

[Weak convergence of probability measures on a separable metric space has several equivalent definitions, one being that for every bounded continuous $g \colon X \to \mathbb{R}$, $\int g \, d\mu_n \to \int g \, d\mu$. The topology of weak convergence is a very nice and physically natural topology. One of its useful properties is that weak convergence can be determined using only countably many bounded continuous functions $g \colon X \to \mathbb{R}$.]

Sorry I don't have a good reference off hand for the above theorem, but I imagine it should be easy to find the result (maybe not stated as one single theorem) in textbooks or lecture notes on descriptive set theory and/or measure theory. The fact that the topology of weak convergence for a Lusin space is Lusin might not be stated explicitly, but what will probably be stated is that the topology of weak convergence for a compact metrisable space is compact; and since $[0,1]$ is obviously compact, combining this statement with all the other parts of the statement of Theorem 1 will then yield that the topology of weak convergence for a Lusin space is Lusin.

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3. Disintegration of measures, and conditional expectation of random measures

For me, one of the marvels of Lusin spaces is the following disintegration theorem. Let $M_X$ be the set of Borel probability measures on $X$.

Theorem 2. Let $(\Omega,\mathcal{F},\mathbb{P})$ be an arbitrary probability space, and let $X$ be a Lusin space. Let $M_{\Omega,X}$ be the set of functions $\dot{\mu}\colon \Omega \to M_X$ such that $\omega \mapsto \dot{\mu}(\omega)(A)$ is an $\mathcal{F}$-measurable map for all $A \in \mathcal{B}(X)$, and let $M_{\Omega,X;\mathbb{P}}$ be the set of equivalence classes of $M_{\Omega,X}$ under the equivalence relation $$ \dot{\mu}_1 \sim \dot{\mu}_2 \quad \Leftrightarrow \quad \textrm{for $\mathbb{P}$-a.a. $\omega \in \Omega$, $\ \dot{\mu}_1(\omega)=\dot{\mu}_2(\omega).$} $$ Now let $M_{\Omega \times X;\mathcal{F},\mathbb{P}}$ be the set of probability measures on the product space $(\Omega \times X, \mathcal{F} \otimes \mathcal{B}(X))$ with the property that $\mu(E \times X)=\mathbb{P}(E)$ for all $E \in \mathcal{F}$. Then $M_{\Omega,X;\mathbb{P}}$ and $M_{\Omega \times X;\mathcal{F},\mathbb{P}}$ are in exact one-to-one correspondence with each other, via the identification $$ \mu(A) \ = \ \int_{\Omega \times X} \mathbf{1}_A(\omega,x) \, \dot{\mu}(\omega)(dx) \, \mathbb{P}(d\omega). $$

I don't know off hand any good textbook for the proof (the famous textbook on random dynamical systems by Ludwig Arnold gives the statement but I think it cites some other book - possibly not in English - for the proof). However, if you can't find the proof easily online, it is proved in my PhD thesis at https://spiral.imperial.ac.uk/handle/10044/1/39569 (Lemma 3.27 / Remark 3.28).

Corollary. For any $\dot{\mu} \in M_{\Omega,X}$ and any sub-$\sigma$-algebra $\mathcal{G}$ of $\mathcal{F}$, there exists an element $\mathbb{E}[\dot{\mu}|\mathcal{G}]$ of $M_{\Omega,X}$ with the property that for all $A \in \mathcal{B}(X)$, the function $\,\omega \mapsto \mathbb{E}[\dot{\mu}|\mathcal{G}](\omega)(A)$ is a version of the conditional expectation $\mathbb{E}[\omega \mapsto \dot{\mu}(\omega)(A)|\mathcal{G}]$.

This is proved by taking the $\mu \in M_{\Omega \times X;\mathcal{F},\mathbb{P}}$ associated to $\dot{\mu}$, and then regarding $\mu$ as an element of $M_{\Omega \times X;\mathcal{G},\mathbb{P}|_\mathcal{G}}$ to go back in the other direction to get $\mathbb{E}[\dot{\mu}|\mathcal{G}]$. (Again, this can be found in Arnold's book.)

Answer 2

Let $D$ be the set of Dirac measures. In fact, $D$ would constitut a reasonable event, regarding probabilistic considerations (since we want to talk about random measures). However, in general it is not an event.

Consider, e.g., $X = [0,1]$ with the cocountable $\sigma$-algebra $\mathcal{B} = \sigma (\{\{x\} : x \in \mathcal{X})$ and note that the $\sigma$-algebra $\mathcal{M}_X$ is countably determined. Then, there is no countable set system $\mathcal{E} \subset \mathcal{B}$ such that $\text{pr}_{\mathcal{E}}^{-1}\big(\text{pr}_{\mathcal{E}}(D)\big) = D$, where $\text{pr}_{\mathcal{E}} : M_X \to [0,\infty]^{\mathcal{E}}$ is the projection of the set functions on $\mathcal{E}.$ This is due to the fact that we can define a measure $\mu$ by $\mu(B) = 1,$ if $B^c$ is countable and else $\mu(B) = 0,$ where $B \in \mathcal{B}.$ Note, $\mu$ is in $\text{pr}_{\mathcal{E}}^{-1}\big(\text{pr}_{\mathcal{E}}(D)\big)$ since $\mu(B) \in \{0,1\}$ for all $B,$ but $\mu$ is not in $D$ since there is no $x\in X$ such that $\mu = \delta_x.$ Hence, $D$ is not $\mathcal{M}_X$-measurable.

Let $(X,\mathcal{T})$ be a locally compact second countable Hausdorff space, then the set of all locally finite measures on $(X,\mathcal{B})$ is $\mathcal{M}_X$-measurable.

To prove this, we first note that there is a countable basis $\mathcal{U}$ of $\mathcal{T}$ and a sequence $(G_k)_{k\in\mathbb{N}}\in\mathcal{T}^\mathbb{N}$ of relatively compact sets such that $G_k \uparrow X$ for $k \to \infty.$ Then, $\mathcal{E} :=\big\{ \bigcap \mathcal{O} : \mathcal{O} \subset \mathcal{U} \cup G(\mathbb{N}),\ |\mathcal{O}| < \infty \big\}$ is a countable and intersection stable generator of $\mathcal{B}.$ According to the measure uniqueness theorem, a locally finite measure $\mu$ is uniquely determined by $\mu|_{\mathcal{E}}$. Consequently, the set of all locally finite measures is countably determined, that is, $\mathcal{M}_X$-measurable.