Aditya Makkar
General Theory of Processes - Part 1
  1. Introduction
  2. Pavings and Mosaics
    1. Properties of pavings and mosaics
    2. Compact pavings
  3. Envelopes
    1. Properties of envelopes
  4. Capacitance
  5. Scrapers
    1. Sierpiński’s Theorem
    2. Mixing of Scrapers
    3. Proof of Sierpiński’s Theorem
  6. Choquet Capacities
  7. Measurable Projection
  8. Measurable Graph
  9. Debut
  10. Measurable Section
  11. Epilogue
  12. References

I have been attending a reading course on stochastic analysis led by Professor Ioannis Karatzas, where the students take turns in presenting a topic of their choice. I recently presented on Choquet's theory of capacities and its applications to measure theory and in the general theory of processes. This blog post is based on this presentation. I have freely copied[1] from many sources, but my primary reference are the (unfortunately unpublished) lecture notes by Prof. Karatzas.

Introduction

Kolmogorov laid the modern axiomatic foundations of probability theory with the German monograph Grundbegriffe der Wahrscheinlichkeitsrechnung which appeared in Ergebnisse Der Mathematik in 1933. This was a period of intense discussions on the foundations of probability, and a majority of probabilists at the time considered measure theory not only a waste of time, but also an offense to "probabilistic intuition" (Meyer, 2009). But by 1950, with the work of Doob in particular, these discussions of foundations had been settled.

Continuous-time processes, on the other hand, were difficult to tame even with measure theory: if a particle is subject to random evolution, to show that its trajectory is continuous, or bounded, requires that all time values be considered, whereas classical measure theory can only handle a countable infinity of time values. Thus, not only does probability depend on measure theory, but it also requires more of measure theory than the rest of analysis (Meyer, 2009).

The missing pieces of the puzzle, which will be the highlight of this and the next blog post, are the debut, section and projection theorems. These theorems are also indispensable in many applications, for instance in dynamic programming and stochastic control (Karoui and Tan, 2013).

To get a taste of these theorems, let's recall a famous error made by Lebesgue in the paper Sur les fonctions représentables analytiquement published in 1905. Consider the measurable space (R2,B(R2))(\mathbb R^2, \mathcal{B}(\mathbb R^2)) and the projection map π\pi given by R2(x,y)π(x,y)=yR.\mathbb R^2 \ni (x,y) \mapsto \pi(x,y) = y \in \mathbb R. It is easy to see that for any open set OO in R2\mathbb R^2, the set π(O)\pi(O) is also open in R\mathbb R: Recall that the standard topology on R2\mathbb R^2 is same as the product topology on R2.\mathbb R^2. By the definition of the product topology on R2\mathbb R^2, an open set OO in R2\mathbb R^2 is of the form O=iIjJiUij×VijO = \bigcup_{i \in I} \bigcap_{j \in J_i} U_{ij} \times V_{ij} for open Uij,VijU_{ij}, V_{ij} in R\mathbb R, II arbitrary and JiJ_i finite. A simple argument gives π(O)=iIjJiUij\pi(O) = \bigcup_{i \in I} \bigcap_{j \in J_i} U_{ij} which is open in R.\mathbb R. In fact, more generally, projection from any product space (with product topology) is an open map. Now it seems reasonable to expect that for any Borel set BB(R2)B \in \mathcal{B}(\mathbb R^2) its projection is also a Borel set in B(R)\mathcal{B}(\mathbb R), and Lebesgue assumed this in his paper. But, in fact, this is FALSE! The error was spotted in around 1917 by Mikhail Suslin, who realised that the projection map need not be Borel, and this lead to his investigation of analytic sets and to begin the study of what is now known as descriptive set theory (Almost Sure blog).

The problem is projection doesn't commute with countable decreasing intersection. For example, consider the decreasing sequence of sets Sn=(0,1/n)×R.S_n = (0, 1/n) \times \mathbb R. Then π(Sn)=R\pi(S_n) = \mathbb R for all nn, giving nNπ(Sn)=R\bigcap_{n \in \mathbb N} \pi(S_n) = \mathbb R, but nNSn=\bigcap_{n \in \mathbb N} S_n = \varnothing, giving π(nNSn)=.\pi \left( \bigcap_{n \in \mathbb N} S_n \right) = \varnothing. The measurable projection theorem stated next will be one of the highlights of this post.

Measurable Projection Theorem: Let (Ω,F,P)(\Omega, \mathcal{F}, \mathbb{P}) be a complete probability space, let (K,B(K))(K, \mathcal{B}(K)) be a locally compact separable metric space endowed with the collection of its Borel sets, and denote by π\pi the projection of K×ΩK \times \Omega onto Ω.\Omega. Then, for every BB(K)FB \in \mathcal{B}(K) \otimes \mathcal{F}, the projection π(B)F.\pi(B) \in \mathcal{F}.

Why is proving such results difficult? As mentioned above it's because projection doesn't behave nicely with intersections. Nevertheless, let us try to see how one might try to prove the above theorem. A standard approach in measure theory is to construct a collection like

E={SK×Ω:π(S)F}\begin{aligned} \mathscr{E} = \{S \subseteq K \times \Omega\,:\,\pi(S) \in \mathcal{F}\}\end{aligned}
which contains the sets satisfying the desired property, and show that it is a σ\sigma-algebra containing a simple collection, say A\mathscr{A}, such that it easy to show that elements of A\mathscr{A} satisfy the desired property and A\mathscr{A} generates B(K)F\mathcal{B}(K) \otimes \mathcal{F}, because then we will have B(K)FE.\mathcal{B}(K) \otimes \mathcal{F} \subseteq \mathscr{E}. To this end, let
A={SK×Ω:S=E×F for EB(K),FF}.\begin{aligned} \mathscr{A} = \{S \subseteq K \times \Omega\,:\, S = E \times F \text{ for } E \in \mathcal{B}(K),\, F \in \mathcal{F}\}.\end{aligned}
Then it is easy to see that AE\mathscr{A} \subseteq \mathscr{E}, A\mathscr{A} generates B(K)F\mathcal{B}(K) \otimes \mathcal{F} and that A\mathscr{A} is an algebra. If we could show that E\mathscr{E} is a monotone class, then we would be done on account of monotone class theorem. Increasing sequences are easily handled, in fact if {Sn}nNK×Ω\{S_n\} _ {n \in \mathbb N} \subseteq K \times \Omega is any sequence, then
π(n=1Sn)=n=1π(Sn).\begin{aligned} \pi\left(\bigcup_{n=1}^\infty S_n\right) = \bigcup_{n=1}^\infty \pi(S_n).\end{aligned}
But if S1S2S_1 \supseteq S_2 \supseteq \cdots, then in general we cannot say
π(n=1Sn)=n=1π(Sn).\begin{aligned} \pi\left(\bigcap_{n=1}^\infty S_n\right) = \bigcap_{n=1}^\infty \pi(S_n).\end{aligned}
Enter Choquet's theory of capacities. It provides the language to prove results like these. We know that every Borel measure μ\mu on Rd\mathbb R^d (in fact, any Polish space) has the interior regularity (also known as tightness) property
μ(B)=supKK(Rd)KBμ(K),  BB(Rd),\begin{aligned} \mu(B) = \sup_{\substack{K \in \mathscr{K}(\mathbb R^d) \\ K \subseteq B}} \mu(K), \quad \forall \; B \in \mathcal{B}(\mathbb R^d),\end{aligned}
where K(Rd)\mathscr{K}(\mathbb R^d) is the collection of all compact sets in Rd.\mathbb R^d. The Choquet's theory of capacities generalizes this approximation-from-below property, and distills those properties of measure that allow for such approximation to hold in very general settings. As we will see, monotonicity and continuity from above and below properties are at play here, and notions corresponding to complements or differences will be absent.

The next few sections will be very abstract and it is easy to lose sight of our goal. Some people enjoy this mental gymnastics, but even if you find this dry, the reward at the end will be worth the initial struggle. We start with Choquet's theory of capacities. The highlight of this part will be Choquet's capacitability theorem. To prove this major result we will need to define a lot of new terminology and prove some major results like Sierpiński's theorem and Sion's theorem. Armed with Choquet's capacitability theorem we will prove Measurable Section theorem which in turn will form the backbone of various other results in measure theory. These results in measure theory will then help us prove results in general theory of processes, but we will discuss this part in the next blog post.

Pavings and Mosaics

Definition 1: Let EE be a nonempty set. A collection E\mathcal E of subsets of EE is called a paving if it closed under finite unions and finite intersections. The pair (E,E)(E, \mathcal E) is then called a paved space.

The concept of paving generalizes the concept of algebra. As an example, if EE is a topological space, then the collection of closed subsets of EE forms a paving. As another example, if EE is a Hausdorff space, then (E,K(E))(E, \mathscr K(E)) is a paved space, where as before K(E)\mathscr{K}(E) denotes the collection of all compact sets in E.E.

It is easy to check that an arbitrary intersection of pavings is also a paving, and that the collection P(E)\mathfrak{P}(E) of all subsets of EE is a paving. Thus for any collection A\mathcal{A} of subsets of EE, we can define the notion of paving generated by A\mathcal{A} as the smallest paving of subsets of EE that contains A\mathcal{A} by simply defining it to be the intersection of all pavings of EE containing A.\mathcal{A}.

Definition 2: For two paved spaces (E,E)(E,\mathcal{E}) and (F,F)(F,\mathcal{F}), the product paving of E\mathcal{E} and F\mathcal{F}, denoted by EpF\mathcal{E} \otimes_p \mathcal{F}, is a paving on E×FE \times F generated by all rectangles R={A×B:AE,BF}.\mathcal{R} = \{A \times B \,:\, A \in \mathcal{E}, B \in \mathcal{F}\}.

Using the fact that (A1×B1)(A2×B2)=(A1A2)×(B1B2)(A_1 \times B_1) \cap (A_2 \times B_2) = (A_1 \cap A_2) \times (B_1 \cap B_2) we see that R\mathcal{R} is stable under finite intersections. Therefore, any element of EpF\mathcal{E} \otimes_p \mathcal{F} is of the form i=1nAi×Bi\bigcup_{i=1}^n A_i \times B_i, where AiEA_i \in \mathcal{E} and BiFB_i \in \mathcal{F} for all i=1,,n.i=1,\ldots,n.

Definition 3: Let EE be a nonempty set. A collection of subsets of EE which is closed under countable unions and countable intersections is called a mosaic.

The concept of mosaic generalizes the concept of σ\sigma-algebra. Just like paving, it is easy to define the notion of mosaic generated by a collection. They will always occur in the context of a paving E\mathcal{E} on E.E. We denote by E^\widehat{\mathcal{E}} the mosaic generated by E.\mathcal{E}. E ^pF\mathcal{E}\ \widehat{\otimes}_p \mathcal{F} will denote the mosaic generated by the product paving EpF.\mathcal{E} \otimes_p \mathcal{F}.

Just like with monotone class arguments in measure theory, the paving E\mathcal E will be a simple collection of subsets of EE for which it is easy to prove a property P.P. From here we will show that the elements of the mosaic E^\widehat{\mathcal E} also satisfy P.P. Often E^\widehat{\mathcal E} will be a σ\sigma-algebra.

Henceforth the notation E\mathcal{E} will be used for a paving on E.E.

Properties of pavings and mosaics

Just like the results connecting algebra and σ\sigma-algebra, we have results connecting pavings and mosaics.

Lemma 1: If AEA \in \mathcal{E} implies Ac=EAE^A^\mathsf{c} = E \setminus A \in \widehat{\mathcal{E}}, then E^=σ(E).\widehat{\mathcal{E}} = \sigma(\mathcal{E}).
Follows immediately from the monotone class theorem.

As an example, if EE is a separable, locally compact metric space, and E=K(E)\mathcal{E} = \mathscr{K}(E) is the collection of compact subsets of EE, then this property holds. In fact, EE can be a second-countable locally compact Hausdorff space. Then it is σ\sigma-compact and every compact subset is closed. This implies that the open set given by the complement of a compact set is a countable union of compact sets. Combine it with the fact that an open subset of a locally compact and σ\sigma-compact is itself σ\sigma-compact to get that if AEA \in \mathcal E, then AcE^.A^\mathsf{c} \in \widehat{\mathcal E}.

Lemma 2: The mosaic E^\widehat{\mathcal{E}} is the smallest collection of subsets of EE that contains E\mathcal{E} and is closed under countable increasing unions and under countable decreasing intersections. In other words, if M(E)\mathcal M(\mathcal{E}) denotes the monotone class generated by E\mathcal{E}, then E^=M(E).\widehat{\mathcal{E}} = \mathcal M(\mathcal{E}).

Since a mosaic is a monotone class and since EE^\mathcal{E} \subseteq \widehat{\mathcal{E}}, we have M(E)E^.\mathcal M(\mathcal{E}) \subseteq \widehat{\mathcal{E}}. For the other side, we will be done if we show that M(E)\mathcal M(\mathcal{E}) is a mosaic, since EM(E).\mathcal{E} \subseteq \mathcal M(\mathcal{E}).

The first property to note is that a monotone paving is a mosaic. To see this, let R\mathcal{R} be a monotone paving and A1,A2,R.A_1, A_2, \ldots \in \mathcal{R}. Then since R\mathcal{R} is a paving, Bn=i=1nAiRB_n = \bigcup_{i=1}^n A_i \in \mathcal{R} for all nN.{n \in \mathbb N}. But {Bn}nN\{B_n\}_{n \in \mathbb N} is an increasing sequence and R\mathcal{R} is a monotone class, and therefore their union nNAn=nNBnR\bigcup_{n \in \mathbb N} A_n = \bigcup_{n \in \mathbb N} B_n \in \mathcal{R}, and hence R\mathcal{R} is closed under countable unions. Similarly for countable intersections.

Therefore, we will be done if we show that M(E)\mathcal M(\mathcal{E}) is a paving. To this end, for any BEB \subseteq E, let

K(B):={AE:AB,ABM(E)}.\begin{aligned} \mathcal{K}(B) := \{A \subseteq E\,:\, A \cup B,\, A \cap B \in \mathcal M(\mathcal{E})\}.\end{aligned}
Notice that by symmetry, AK(B)    BK(A).A \in \mathcal{K}(B) \iff B \in \mathcal{K}(A). If A1A2K(B)A_1 \subseteq A_2 \subseteq \cdots \in \mathcal{K}(B) is an increasing sequence, then
nNAnB=nN(AnB)M(E)nNAnB=nN(AnB)M(E),\begin{aligned} \bigcup_{n \in \mathbb N} A_n \cup B &= \bigcup_{n \in \mathbb N} (A_n \cup B) \in \mathcal M(\mathcal{E}) \\ \bigcup_{n \in \mathbb N} A_n \cap B &= \bigcup_{n \in \mathbb N} (A_n \cap B) \in \mathcal M(\mathcal{E}),\end{aligned}
and similarly for decreasing sequences. Therefore, if K(B)\mathcal{K}(B) \neq \varnothing, then it is a monotone class. If A,BEA, B \in \mathcal{E}, then by the definition of a paving, AK(B).A \in \mathcal{K}(B). Since this is true for every AEA \in \mathcal{E}, we have EK(B)\mathcal{E} \subseteq \mathcal{K}(B), and K(B)\mathcal{K}(B) is a monotone class containing E.\mathcal{E}. Since M(E)\mathcal M(\mathcal{E}) is the smallest monotone class containing E\mathcal{E}, we have
M(E)K(B).\begin{aligned} \mathcal M(\mathcal{E}) \subseteq \mathcal{K}(B).\end{aligned}
Hence if AM(E)A \in \mathcal M(\mathcal{E}) and BEB \in \mathcal{E}, then AK(B)A \in \mathcal{K}(B), and therefore BK(A).B \in \mathcal{K}(A). Since this is true for every BEB \in \mathcal{E}, it follows that
M(E)K(A).\begin{aligned} \mathcal M(\mathcal{E}) \subseteq \mathcal{K}(A).\end{aligned}
The validity of this relation for every AM(E)A \in \mathcal M(\mathcal{E}) is equivalent to the assertion that M(E)\mathcal M(\mathcal{E}) is a paving.

Compact pavings

Recall a standard result from topology: a topological space XX is compact if and only if for every collection C\mathcal C of closed subsets of XX having the finite intersection property (i.e., every finite sub-collection of C\mathcal C has nonempty intersection), the intersection CCC\bigcap_{C \in \mathcal C} C is nonempty. For our use case, we consider countable sub-collections.

Definition 4: Consider an arbitrary collection A\mathcal A of subsets of E.E. It is called a compact collection if every countable sub-collection of elements in A\mathcal A having the finite intersection property has non-empty intersection. A paving E\mathcal{E} is a compact paving, if every decreasing sequence of nonempty elements of E\mathcal{E} has a nonempty intersection.

It is easy to verify that a compact paving is a compact collection. As an example, if EE is a separable metric space, the collection, K(E)\mathscr{K}(E), of all compact subsets of EE is a compact paving. This follows immediately from the finite intersection characterization of a compact topological space stated above.

We define

Eδ:={nNAn:AnE for all nN}\begin{aligned} \mathcal{E}_\delta := \left\{\bigcap_{n \in \mathbb N} A_n\,:\,A_n \in \mathcal{E} \text{ for all } n \in \mathbb N\right\}\end{aligned}
and note that if E\mathcal{E} is a compact paving then so is Eδ.\mathcal{E}_\delta. Indeed, Eδ\mathcal{E}_\delta is a paving because for A=nNAnA = \bigcap_{n \in \mathbb N} A_n and B=mNBmB = \bigcap_{m \in \mathbb N} B_m in Eδ\mathcal{E}_\delta,
AB=nNmN(AnBm)EδAB=nNmN(AnBm)Eδ,\begin{aligned} A \cup B &= \bigcap_{n \in \mathbb N} \bigcap_{m \in \mathbb N} (A_n \cup B_m) \in \mathcal{E}_\delta \\ A \cap B &= \bigcap_{n \in \mathbb N} \bigcap_{m \in \mathbb N} (A_n \cap B_m) \in \mathcal{E}_\delta,\end{aligned}
and Eδ\mathcal{E}_\delta is a compact paving because intersection of a sequence of countable intersections is again a countable intersection.

The next lemma tells us when it acceptable to commute projection with countable intersections.

Lemma 3: Let KK and EE be two nonempty sets, and denote by π\pi the projection of K×EK \times E onto E.E. Suppose that H\mathcal{H} is a paving of subsets of K×EK \times E with the property that, for every xEx \in E, the collection H(x):={H(x):HH}\mathcal{H}(x) := \{H(x) \,:\, H \in \mathcal{H}\} is a compact paving on K.K. Here H(x):={yK:(y,x)H}H(x) := \{y \in K \,:\, (y,x) \in H\} denotes the xx-section of HK×E.H \subseteq K \times E. Then, for every decreasing sequence {Hn}nN\{H_n\} _ {n \in \mathbb N} of sets in the paving Hδ\mathcal{H}_\delta, we have
π(nNHn)=nNπ(Hn).\begin{aligned} \pi\left(\bigcap _ {n \in \mathbb N} H_n\right) = \bigcap _ {n \in \mathbb N} \pi(H_n).\end{aligned}

π(nNHn)nNπ(Hn)\pi\left(\bigcap_{n \in \mathbb N} H_n\right) \subseteq \bigcap_{n \in \mathbb N} \pi(H_n) is easy to see because if xπ(nNHn)x \in \pi\left(\bigcap_{n \in \mathbb N} H_n\right) then there exists (y,x)K×E(y,x) \in K \times E such that (y,x)nNHn(y,x) \in \bigcap_{n \in \mathbb N} H_n which implies xπ(Hn)x \in \pi(H_n) for all nN.n \in \mathbb N.

For the other side, let xnNπ(Hn).x \in \bigcap_{n \in \mathbb N} \pi(H_n). Then the sequence {Hn(x)}nN\{H_n(x)\}_{n \in \mathbb N} is decreasing whose elements are nonempty and they are in H(x)δ.\mathcal{H}(x)_\delta. But since H(x)δ\mathcal{H}(x)_\delta is a compact paving by assumption, nHn(x)\bigcap_n H_n(x) must be nonempty, implying xπ(nNHn).x \in \pi\left(\bigcap_{n \in \mathbb N} H_n\right).

Envelopes

Definition 5: Let (E,E)(E, \mathcal{E}) be a paved space, and fix a subset AEA \subseteq E as well as a decreasing sequence {Ak}kNP(E).\{A_k\} _ {k \in \mathbb N} \subseteq \mathfrak{P}(E). We say that AA is an E\mathcal{E}-envelope of {Ak}kN\{A_k\} _ {k \in \mathbb N}, if there exists a decreasing sequence {Ck}kNE{E}\{C_k\}_{k \in \mathbb N} \subseteq \mathcal{E} \cup \{E\} such that AkCk,    kN and kNCkA.\begin{aligned} A_k \subseteq C_k, \; \forall \; k \in \mathbb N \quad \text{ and } \quad \bigcap _ {k \in \mathbb N} C_k \subseteq A.\end{aligned}

Examples:

  1. Let EE be a separable metric space, and E\mathcal{E} be the paving consisting of all closed subsets of E.E. Then a subset AA of EE is an E\mathcal{E}-envelope of a given decreasing sequence {Ak}kNP(E)\{A_k\} _ {k \in \mathbb N} \subseteq \mathfrak{P}(E) if, and only if, AA contains kNAk\bigcap_{k \in \mathbb N} \overline{A}_k, the intersection of the closures of the sets Ak,kNA_k, k \in \mathbb N in the sequence. Indeed, that AA is an E\mathcal{E}-envelope of {Ak}kN\{A_k\} _ {k \in \mathbb N} (with the existence of a sequence {Ck}kN\{C_k\}_{k \in \mathbb N} as in Definition 5 above) implies AA contains kNAk\bigcap_{k \in \mathbb N} \overline{A}_k follows from the observation that AkCk\overline{A}_k \subseteq C_k for each kNk \in \mathbb N, while the other side is easy to see if we let Ck=AkC_k = \overline{A}_k for each kN.k \in \mathbb N.

  2. An abstract version of the example above: Let (E,E)(E, \mathcal{E}) be a paved space; for every subset AA of EE, introduce the collection of sets A:={BE{E}:AB}\mathcal{A} := \{B \in \mathcal{E} \cup \{E\} \,:\, A \subseteq B\} and assume that the intersection A:=BAB\overline{A} := \bigcap_{B \in \mathcal{A}} B, called the adherent of AA in the paving E\mathcal{E}, belongs to Eδ{E}\mathcal{E}_\delta \cup \{E\}, i.e., A\overline{A} is a countable intersection of sets in E{E}.\mathcal{E} \cup \{E\}. We claim, and show next, that AA is an E\mathcal{E}-envelope of a given decreasing sequence {Ak}kNP(E)\{A_k\}_{k \in \mathbb N} \subseteq \mathfrak{P}(E) if, and only if, AA contains kNAk.\bigcap_{k \in \mathbb N} \overline{A}_k.

Lemma 4: In the setting of the last example, a subset AA of EE is an E\mathcal{E}-envelope of a given decreasing sequence {Ak}kNP(E)\{A_k\} _ {k \in \mathbb N} \subseteq \mathfrak{P}(E) if, and only if, AA contains kNAk.\bigcap_{k \in \mathbb N} \overline{A}_k.

The necessity is clear: if there exists a decreasing sequence {Ck}kNE{E}\{C_k\}_{k \in \mathbb N} \subseteq \mathcal{E} \cup \{E\} such that (Env)(\text{Env}) is satisfied then AkCk\overline{A}_k \subseteq C_k and therefore kNAkkNCkA.\bigcap_{k \in \mathbb N} \overline{A}_k \subseteq \bigcap_{k \in \mathbb N} C_k \subseteq A.

To see the sufficiency, for every kNk \in \mathbb N, let {Bnk}nNE{E}\left\{B_n^k\right\}_{n \in \mathbb N} \subseteq \mathcal{E} \cup \{E\} be a decreasing sequence such that Ak=nNBnk\overline{A}_k = \bigcap_{n \in \mathbb N} B_n^k (such a sequence exists because of the assumption in Example 2). Then

Ck:=Bk1Bk2Bkk,kN\begin{aligned} C_k := B_k^1 \cap B_k^2 \cap \cdots \cap B_k^k, \quad k \in \mathbb N\end{aligned}
defines a decreasing sequence of elements in E{E}\mathcal{E} \cup \{E\} with AkAkCkA_k \subseteq \overline{A}_k \subseteq C_k and kNAk=kNCk.\bigcap_{k \in \mathbb N} \overline{A}_k =\bigcap_{k \in \mathbb N} C_k. It follows that the set AA envelops the sequence {Ak}kN\{A_k\}_{k \in \mathbb N}, if AA contains the countable intersection kNAk\bigcap_{k \in \mathbb N} \overline{A}_k, for then the decreasing sequence {Ck}kNE{E}\{C_k\}_{k \in \mathbb N} \subseteq \mathcal{E} \cup \{E\} satisfies the requirements of (1).

Properties of envelopes

The next lemma lists some properties of envelopes which we will be using frequently.

Lemma 5:

  1. If AA is an envelope of a given decreasing sequence {An}nNP(E)\{A_n\} _ {n \in \mathbb N} \subseteq \mathfrak{P}(E), then every subset of EE that contains AA is also an envelope of {An}nN.\{A_n\} _ {n \in \mathbb N}.

  2. Two decreasing sequences of subsets of EE that possess a common subsequence, admit the exact same envelopes.

  3. The collection of envelopes of a given decreasing sequence of subsets of EE, is closed under countable intersections.

Parts 1. and 2. are trivial, and follow immediately from the definition.

For part 3., let {Ak}kN\{A^k\} _ {k \in \mathbb N} be a sequence of envelopes of a given decreasing sequence {An}nNP(E).\{A_n\} _ {n \in \mathbb N} \subseteq \mathfrak{P}(E). For each kNk \in \mathbb N, let {Bnk}nNE{E}\{B_n^k\} _ {n \in \mathbb N} \subseteq \mathcal{E} \cup \{E\} be a decreasing sequence, such that AnBnkA_n \subseteq B_n^k for all nNn \in \mathbb N and nNBnkAk.\bigcap_{n \in \mathbb N} B_n^k \subseteq A^k. Then

Cn:=Bn1Bn2Bnn,nN\begin{aligned} C_n := B_n^1 \cap B_n^2 \cap \cdots \cap B_n^n, \quad n \in \mathbb N\end{aligned}
defines a decreasing sequence of elements in E{E}\mathcal{E} \cup \{E\} that satisfies AnCnA_n \subseteq C_n and nNCn=(k,n)N2BnkkNAk.\bigcap_{n \in \mathbb N} C_n = \bigcap_{(k,n) \in \mathbb N^2} B_n^k \subseteq \bigcap_{k \in \mathbb N} A^k. It follows that kNAk\bigcap_{k \in \mathbb N} A^k is an E\mathcal{E}-envelope of {An}nN.\{A_n\}_{n \in \mathbb N}.

Capacitance

Definition 6: Let EE be a nonempty set. A collection C\mathcal C of subsets of EE is called a capacitance, if

  1. whenever ACA \in \mathcal{C} and ABA \subseteq B, then BCB \in \mathcal{C}, and

  2. whenever {An}nN\{A_n\} _ {n \in \mathbb N} is an increasing sequence of subsets of EE such that nNAnC\bigcup_{n \in \mathbb N} A_n \in \mathcal{C}, there is an integer mm such that AmC.A_m \in \mathcal{C}.

Intuitively, a capacitance is a collection of “big” sets: the power set P(E)\mathfrak{P}(E) is a capacitance, and so are the collections of nonempty and of uncountable subsets of E.E. The notion of pre-capacity, defined next, gives a more useful example.

Definition 7: A function I ⁣:P(E)RI \colon \mathfrak{P}(E) \to \overline{\mathbb R} is called a pre-capacity, if it is

  1. monotone increasing, i.e., I(A)I(B)I(A) \le I(B) holds for every ABA \subseteq B, and

  2. ascending, i.e., for every increasing sequence {An}nN\{A_n\} _ {n \in \mathbb N} we have

I(nNAn)=supnNI(An).\begin{aligned} I\left(\bigcup_{n \in \mathbb N} A_n\right) = \sup_{n \in \mathbb N} I(A_n).\end{aligned}

If I ⁣:P(E)RI \colon \mathfrak{P}(E) \to \overline{\mathbb R} is a pre-capacity, then for every given real number tt the collection

C={AP(E):I(A)>t}\begin{aligned} \mathcal{C} = \{A \in \mathfrak{P}(E) \,:\, I(A) > t\}\end{aligned}
is a capacitance. Conversely, given a capacitance C\mathcal{C}, one can associate to it a pre-capacity by defining
I(A)={1if AC0if AC.\begin{aligned} I(A) = \begin{cases} 1 &\text{if } A \in \mathcal{C} \\ 0 &\text{if } A \notin \mathcal{C}. \end{cases}\end{aligned}
This then leads to the identification C={AP(E):I(A)>0}.\mathcal{C} = \{A \in \mathfrak{P}(E) \,:\, I(A) > 0\}.

Henceforth assume that there is an underlying paved space (E,E)(E, \mathcal{E}) and a capacitance C\mathcal{C} of subsets of E.E.

Scrapers

missing
Scraping or planing

Definition 8: A sequence f={fn}nN\mathfrak{f} = \{f_n\} _ {n \in \mathbb N} of mappings fn ⁣:(P(E))nP(E)f_n \colon \left(\mathfrak{P}(E)\right)^n \to \mathfrak{P}(E) is called a Sierpiński’s C\mathcal{C}-scraper, or simply a C\mathcal{C}-scraper, if

  1. fn(B1,B2,,Bn)Bnf_n(B_1, B_2, \ldots, B_n) \subseteq B_n for all nNn \in \mathbb N and for all sets B1,,BnP(E)B_1, \ldots, B_n \in \mathfrak{P}(E), and

  2. whenever BnCB_n \in \mathcal{C}, then fn(B1,B2,,Bn)C.f_n(B_1, B_2, \ldots, B_n) \in \mathcal{C}.

The first property expresses the intuitive notion that fn(B1,B2,,Bn)f_n(B_1, B_2, \ldots, B_n) “scrapes” BnB_n and the second property ensures that “the scraping does not remove too big a chunk” from Bn.B_n. In French, scraper is called rabotage which can be translated also as planing. A simple example of a scraper is the identity scraper: fn(B1,Bn)=Bnf_n(B_1, \ldots B_n) = B_n for all nNn \in \mathbb N and for all sets B1,,BnP(E).B_1, \ldots, B_n \in \mathfrak{P}(E).

Definition 9: Given a C\mathcal{C}-scraper f={fn}nN\mathfrak{f} = \{f_n\} _ {n \in \mathbb N}, a (necessarily decreasing) sequence {Bn}nN\{B_n\} _ {n \in \mathbb N} of subsets of EE will be called f\mathfrak{f}-scraped, if for all nNn \in \mathbb N we have
Bn+1fn(B1,B2,,Bn)andBnC.\begin{aligned} B_{n+1} \subseteq f_n(B_1, B_2, \ldots, B_n) \quad \text{and} \quad B_n \in \mathcal{C}.\end{aligned}
Definition 10: For any BP(E)B \in \mathfrak{P}(E) and C\mathcal{C}-scraper f={fn}nN\mathfrak{f} = \{f_n\} _ {n \in \mathbb N}, the sequence {Pn}nNC\{P_n\} _ {n \in \mathbb N} \subseteq \mathcal{C} defined by P1:=BP_1 := B and Pn+1:=fn(P1,,Pn)P_{n+1} := f_n(P_1, \ldots, P_n) for all nNn \in \mathbb N is f\mathfrak{f}-scraped. It is called the f\mathfrak{f}-scraped orbit of BB.

(Dellacherie, 1972) calls {Pn}nN\{P_n\} _ {n \in \mathbb N} above as the f\mathfrak f-scraped sequence deduced from B.B.

Definition 11: A C\mathcal{C}-scraper f={fn}nN\mathfrak{f} = \{f_n\}_{n \in \mathbb N} is called compatible with a given set AP(E)A \in \mathfrak{P}(E), if AA envelopes every f\mathfrak{f}-scraped sequence {Bn}nN\{B_n\} _ {n \in \mathbb N} with B1A.B_1 \subseteq A.

A set AP(E)A \in \mathfrak{P}(E) is smooth for the capacitance C\mathcal{C}, if there exists a C\mathcal{C}-scraper compatible with it.

If ACA \notin \mathcal C, then no subset of AA can be in C\mathcal C either by the definition of a capacitance. This implies that there does not exist a f\mathfrak f-scraped sequence {Bn}nN\{B_n\}_{n \in \mathbb N} satisfying B1A.B_1 \subset A. On the other hand, with the identity scraper f\mathfrak f, AA envelopes every f\mathfrak f-scraped sequence {Bn}nN\{B_n\}_{n \in \mathbb N} because B1B_1 must equal A.A. Thus AA is smooth.

If ACA \in \mathcal C and is smooth, then there always exists a sequence {Bn}nN\{B_n\}_{n \in \mathbb N} of which AA is an envelope. Indeed, by assumption there exists f\mathfrak f, a scraper, compatible with A.A. Let {Bn}nN\{B_n\}_{n \in \mathbb N} be the f\mathfrak f-scraped orbit of A.A. Then since AA is smooth, AA envelops {Bn}nN.\{B_n\}_{n \in \mathbb N}.

Sierpiński’s Theorem

The next result is central in this theory.

Theorem 1 [Sierpiński]: Let (E,E)(E, \mathcal{E}) be a paved space, and C\mathcal{C} a capacitance. The collection of subsets of EE which are smooth for the capacitance, is closed under countable increasing unions and under countable intersections.

We will come back to its proof later. Let's prove some of its useful consequences first.

Theorem 2: Let (E,E)(E ,\mathcal{E}) be a paved space, and C\mathcal{C} a capacitance. The elements of the mosaic E^\widehat{\mathcal{E}} generated by E\mathcal{E} are smooth.
An easy consequence of Theorem 1, Lemma 2 and the fact that every element of E\mathcal{E} is smooth because they are compatible with the identity scraper. \square

This theorem is very useful in proving some important results. We will discuss two of them. The first one is the metric space version of Choquet's capacitability theorem. The proof of it will be very similar to the general Choquet's capacitability theorem, but we will need Sion's theorem for the general version, which will be the second result.

Definition 12: Let EE be a compact metric space, endowed with the paving E=K(E)\mathcal{E} = \mathscr{K}(E) of its compact sets. II is called a metric capacity on (E,E)(E, \mathcal{E}) if it a pre-capacity that "descends on compacts" in the sense that for every decreasing sequence {Kn}nNK(E)\{K_n\} _ {n \in \mathbb N} \subseteq \mathscr{K}(E) it satisfies
I(nNKn)=infnNI(Kn).\begin{aligned} I\left(\bigcap _ {n \in \mathbb N}K_n\right) = \inf _ {n \in \mathbb N} I(K_n).\end{aligned}
Theorem 3 [Metric space version of Choquet's capacitability theorem]: For every Borel subset BB of a compact metric space EE, and any metric capacity I ⁣:P(E)RI \colon \mathfrak{P}(E) \to \overline{\mathbb R}, we have
I(B)=supKK(E)KBI(K).\begin{aligned} I(B) = \sup_{\substack{K \in \mathscr{K}(E) \\ K \subseteq B}} I(K).\end{aligned}

Fix an arbitrary BB(E).B \in \mathcal{B}(E). If I(B)=I(B) = -\infty then I(K)=I(K) = -\infty for any KBK \subseteq B, and we have our desired equality trivially. Otherwise, we need to show that whenever I(B)>tI(B) > t holds for some given real number tt, there exists a compact set KBK \subseteq B such that I(K)t.I(K) \ge t. Recall that

C={AP(E):I(A)>t}\begin{aligned} \mathcal{C} = \{A \in \mathfrak{P}(E)\,:\,I(A) > t\}\end{aligned}
is a capacitance. Also recall that a subset AA of EE is an E\mathcal{E}-envelope of a decreasing sequence {An}nNP(E)\{A_n\}_{n \in \mathbb N} \subseteq \mathfrak{P}(E) if and only if AA contains nNAn.\bigcap_{n \in \mathbb N}\overline{A}_n.

Lemma 1 gives that the mosaic E^\widehat{\mathcal{E}} generated by E=K(E)\mathcal{E} = \mathscr{K}(E) coincides with the Borel σ\sigma-algebra B(E).\mathcal{B}(E). Hence by Theorem 2 every Borel set is smooth. Thus, there exists a C\mathcal{C}-scraper f={fn}nN\mathfrak{f} = \{f_n\}_{n \in \mathbb N} compatible with the set B.B.

Consider the f\mathfrak{f}-scraped orbit {Pn}nNC\{P_n\} _ {n \in \mathbb N} \subseteq \mathcal{C} of B.B. By construction BB is an envelope of {Pn}nN\{P_n\} _ {n \in \mathbb N}, and hence it contains K:=nNPn.K := \bigcap _ {n \in \mathbb N} \overline{P}_n. KK is closed and hence also compact on account of being a subset of a compact set EE; similarly for Pn\overline{P}_n for all nN.n \in \mathbb N. But since {Pn}nNC\{P_n\} _ {n \in \mathbb N} \subseteq \mathcal{C} we have I(Pn)>tI(\overline{P}_n) > t for all nN.n \in \mathbb N. Now use the descending on compacts property of II to get

I(K)=I(nNPn)=infnNI(Pn)t.\begin{aligned} I(K) = I\left(\bigcap _ {n \in \mathbb N} \overline{P}_n\right) = \inf _ {n \in \mathbb N} I(\overline{P}_n) \ge t.\end{aligned}

Theorem 4 [Sion's theorem]: Let (E,E)(E,\mathcal{E}) be a paved space, and C\mathcal{C} a capacitance. For every element BB of CE^\mathcal{C} \cap \widehat{\mathcal{E}}, there exists a decreasing sequence {Kn}nNCE\{K_n\} _ {n \in \mathbb N} \subseteq \mathcal{C} \cap \mathcal{E} such that nNKnB.\bigcap _ {n \in \mathbb N} K_n \subseteq B.

Theorem 2 implies that the set BB is smooth, and thus there exists a C\mathcal{C}-scraper f={fn}nN\mathfrak{f} = \{f_n\} _ {n \in \mathbb N} compatible with it. Let {Pn}nNC\{P_n\} _ {n \in \mathbb N} \subseteq \mathcal{C} be the f\mathfrak{f}-scraped orbit of B.B. Then BB is an envelope of {Pn}nN\{P_n\} _ {n \in \mathbb N}, so there exists a decreasing sequence {Bn}nN\{B_n\} _ {n \in \mathbb N} of subsets of E{E}\mathcal{E} \cup \{E\} such that nNBnP(B)\bigcap _ {n \in \mathbb N} B_n \subseteq \mathfrak{P}(B) and PnBnP_n \subseteq B_n for all nN.{n \in \mathbb N}. Notice BnC.B_n \in \mathcal{C}.

If the sets BnB_n belong to the paving E\mathcal{E} from a certain index mm onward, we take Kn:=Bm+n,nNK_n := B_{m+n}, n \in \mathbb N as our sequence. Otherwise if Bn=EB_n = E holds for all integers nn, the set B=EB=E is the union of an increasing sequence of sets in E\mathcal{E} because BE^B \in \widehat{\mathcal{E}} and by Lemma 2. Therefore, the fact that BCB \in \mathcal{C} implies BB contains a set KCEK \in \mathcal{C} \cap \mathcal{E}; it then suffices to take Kn=KK_n = K for all integers n.n.

We now come back to the proof of Theorem 1. But first we will need the following clever operation on scrapers, and a couple of results.

Mixing of Scrapers

Definition 13: Consider a sequence {fk,kN}={{fnk}nN,kN}\left\{\mathfrak{f}^k, k \in \mathbb N\right\} = \left\{\{f^k_n\} _ {n \in \mathbb N}, k \in \mathbb N\right\} of scrapers, and a bijection
N2(p,q)β(p,q)=pqN,\begin{aligned} \mathbb N^2 \ni (p,q) \mapsto \beta(p,q) = p \star q \in \mathbb N,\end{aligned}
which is strictly increasing in each of its arguments. For every integer nN{n \in \mathbb N} and sets P1,P2,,PnP_1, P_2, \ldots, P_n, if n=pqn = p \star q, let
fn(P1,P2,,Pn):=fqp(Pp1,Pp2,,Ppq).\begin{aligned} f_n(P_1, P_2, \ldots, P_n) := f^p_q(P_{p \star 1}, P_{p \star 2}, \ldots, P_{p \star q}).\end{aligned}
It is easy to see that this defines a new scraper f={fn}nN\mathfrak{f} = \{f_n\}_{n \in \mathbb N}, called the mixing of the scrapers {fk,kN}\left\{\mathfrak{f}^k, k \in \mathbb N\right\} via the bijection β\beta.
Theorem 5: Let {fk,kN}\left\{\mathfrak{f}^k, k \in \mathbb N\right\} be a sequence of scrapers, and denote by f\mathfrak{f} its mixing by a bijection β.\beta. In order for a subset AA of EE to be compatible with f\mathfrak{f}, it suffices that it be compatible with one of the scrapers fk,kN.\mathfrak{f}^k, k \in \mathbb N.

Let AP(E)A \in \mathfrak{P}(E) be compatible with fk\mathfrak{f}^k for some arbitrary but fixed k.k. Consider also a sequence of sets {Pn}nN\{P_n\} _ {n \in \mathbb N}, which is f\mathfrak{f}-scraped and whose first term P1P_1 is contained in A.A. We need to show that the set AA envelops {Pn}nN.\{P_n\} _ {n \in \mathbb N}.

To do this, we exploit Lemma 5 (ii) and construct a decreasing sequence {Qn}nNP(E)\{Q_n\} _ {n \in \mathbb N} \subseteq \mathfrak{P}(E) which is a subsequence of {Pn}nN\{P_n\} _ {n \in \mathbb N} and show that AA envelops {Qn}nN.\{Q_n\} _ {n \in \mathbb N}. This will then imply AA envelops {Pn}nN.\{P_n\} _ {n \in \mathbb N}. To this end, define

Qn:=Pkn  nN.\begin{aligned} Q_n := P_{k \star n} \quad \forall \; {n \in \mathbb N}.\end{aligned}
Because Q1=Pk1P11=P1AQ_1 = P_{k \star 1} \subseteq P_{1 \star 1} = P_1 \subseteq A and AA is compatible with fk\mathfrak{f}^k, to show that AA envelops {Qn}nN\{Q_n\}_{n \in \mathbb N} it suffices to show {Qn}nN\{Q_n\} _ {n \in \mathbb N} is fk\mathfrak{f}^k-scraped.

Now, Qn=PknCQ_n = P_{k \star n} \in \mathcal{C} for all nN{n \in \mathbb N}, so all that remains to be shown is that Qn+1fnk(Q1,Q2,,Qn)Q_{n+1} \subseteq f^k_n(Q_1, Q_2, \ldots, Q_n) holds for all nN.{n \in \mathbb N}. Because {Pn}nN\{P_n\} _ {n \in \mathbb N} is f\mathfrak{f}-scraped we have

Qn+1=Pk(n+1)P1+knfkn(P1,P2,,Pkn)=fnk(Q1,Q2,,Qn),\begin{aligned} Q_{n+1} = P_{k \star (n+1)} \subseteq P_{1 + k\star n} \subseteq f_{k \star n}(P_1, P_2, \ldots, P_{k \star n}) = f^k_n(Q_1, Q_2, \ldots, Q_n),\end{aligned}
giving the desired result.

An immediate corollary of this theorem: If {An}nN\{A_n\}_{n \in \mathbb N} is a sequence of smooth subsets of EE, there exists a scraper f\mathfrak{f} which is compatible with all the sets An,nN.A_n, {n \in \mathbb N}.

Proof of Sierpiński’s Theorem

We state Theorem 1 again:

Theorem 1 [Sierpiński]: Let (E,E)(E, \mathcal{E}) be a paved space, and C\mathcal{C} a capacitance. The collection of subsets of EE which are smooth for the capacitance, is closed under countable increasing unions and under countable intersections.

Closure under countable intersections:

Suppose {Ak}kN\left\{A^k\right\} _ {k \in \mathbb N} is a sequence of smooth sets, A=kNAkA = \bigcap _ {k \in \mathbb N} A^k, and f\mathfrak{f} is a C\mathcal{C}-scraper compatible with all of the sets Ak,kN.A^k, k \in \mathbb N. If {Pn}nN\{P_n\} _ {n \in \mathbb N} is an f\mathfrak{f}-scraped sequence of sets such that P1AP_1 \subseteq A, then P1AkP_1 \subseteq A^k for all kN.k \in \mathbb N. Our construction then implies AkA^k is an E\mathcal{E}-envelope of {Pn}nN\{P_n\} _ {n \in \mathbb N} for all kN.k \in \mathbb N. Lemma 5 (iii) now implies AA is also an E\mathcal{E}-envelope {Pn}nN\{P_n\} _ {n \in \mathbb N}, showing that AA is compatible with f\mathfrak{f}, and hence smooth.

Closure under countable increasing unions:

Suppose {Ak}kN\left\{A^k\right\}_{k \in \mathbb N} is an increasing sequence of smooth sets, A=kNAkA = \bigcup _ {k \in \mathbb N} A^k, and f\mathfrak{f} is a C\mathcal{C}-scraper compatible with all of the sets Ak,kN.A^k, k \in \mathbb N. The scraper f\mathfrak{f} doesn't work for this case and so we create a new one. For any nNn \in \mathbb N and sets P1,P2,,PnP_1, P_2, \ldots, P_n we define

φn(P1,P2,,Pn)={Pnif AP1Cfn(ApP1,P2,,Pn)if AP1C,\begin{aligned} \varphi_n(P_1, P_2, \ldots, P_n) = \begin{cases} P_n &\text{if } A \cap P_1 \notin \mathcal{C} \\ f_n(A^p \cap P_1, P_2, \ldots, P_n) &\text{if } A \cap P_1 \in \mathcal{C}, \end{cases}\end{aligned}
where pp is the smallest integer such that ApP1C.A^p \cap P_1 \in \mathcal{C}. Such an integer does exist from part 2. of the definition of capacitance with the sequence being AkP1AP1C.A^k \cap P_1 \uparrow A \cap P_1 \in \mathcal C. It is easy to see that Φ={φn}nN\Phi = \{\varphi_n\} _ {n \in \mathbb N} is a C\mathcal{C}-scraper. It is sufficient to show that Φ\Phi is compatible with AA to finish our proof.

Let {Pn}nN\{P_n\} _ {n \in \mathbb N} be a Φ\Phi-scraped sequence of sets such that P1A.P_1 \subseteq A. By definition P1CP_1 \in \mathcal{C} and AP1=P1A \cap P_1 = P_1, and so from our construction φn(P1,P2,,Pn)=fn(ApP1,P2,,Pn).\varphi_n(P_1, P_2, \ldots, P_n) = f_n(A^p \cap P_1, P_2, \ldots, P_n). All elements of the sequence ApP1,P2,,Pn,A^p \cap P_1, P_2, \ldots, P_n, \ldots are in C\mathcal{C} and for all nNn \in \mathbb N

Pn+1φn(P1,P2,,Pn)=fn(ApP1,P2,,Pn),\begin{aligned} P_{n+1} \subseteq \varphi_n(P_1, P_2, \ldots, P_n) = f_n(A^p \cap P_1, P_2, \ldots, P_n),\end{aligned}
and thus it follows that this sequence is f\mathfrak{f}-scraped. Now since ApA^p is compatible with f\mathfrak{f}, ApA^p is an envelope of this sequence, and also of {Pn}nN\{P_n\} _ {n \in \mathbb N} by Lemma 5 (ii). It follows that AA is an envelope of {Pn}nN\{P_n\} _ {n \in \mathbb N} by Lemma 5 (i) because ApA.A^p \subseteq A.

Choquet Capacities

Definition 14: A mapping I ⁣:P(E)RI \colon \mathfrak{P}(E) \to \overline{\mathbb R} is called a Choquet E\mathcal{E}-capacity, or simply E\mathcal{E}-capacity, if it is

  1. monotone increasing, i.e., I(A)I(B)I(A) \le I(B) holds for every ABA \subseteq B,

  2. ascending, i.e., for every increasing sequence {An}nNP(E)\{A_n\} _ {n \in \mathbb N} \subseteq \mathfrak{P}(E) we have

I(nNAn)=supnNI(An),\begin{aligned} I\left(\bigcup _ {n \in \mathbb N} A_n\right) = \sup _ {n \in \mathbb N} I(A_n),\end{aligned}
  1. descending on pavings, i.e., for every decreasing sequence {En}nNE\{E_n\} _ {n \in \mathbb N} \subseteq \mathcal{E} we have

I(nNEn)=infnNI(En).\begin{aligned} I\left(\bigcap _ {n \in \mathbb N} E_n\right) = \inf _ {n \in \mathbb N} I(E_n).\end{aligned}
Definition 15: A set AP(E)A \in \mathfrak{P}(E) is called II-capacitable if
I(A)=supKEδKAI(K).\begin{aligned} I(A) = \sup_{\substack{K \in \mathcal{E}_\delta \\ K \subseteq A}} I(K).\end{aligned}

Examples:

  1. Consider a paved space (E,E)(E, \mathcal{E}), where E\mathcal E is a compact paving, and define I(A)=0I(A) = 0 if A=A = \varnothing, I(A)=1I(A) = 1 if A.A \neq \varnothing. Then II is a Choquet E\mathcal{E}-capacity. The property 3. in the definition of a capacity reflects now the assumption that the paving E\mathcal{E} is compact.

  2. Consider a probability space (Ω,F,P)(\Omega, \mathcal{F}, \mathbb{P}), then the outer measure

    P(A):=infBFABP(B),\begin{aligned} \mathbb{P}^*(A) := \inf_{\substack{B \in \mathcal{F} \\ A \subseteq B}} \mathbb{P}(B),\end{aligned}
    which is well-known to be a monotone, countably sub-additive set function taking the value 00 on the empty set \varnothing, is a Choquet F\mathcal{F}-capacity. Proof of this is a standard exercise in measure theory, albeit with a different terminology of continuity from below.

  3. Consider a locally compact, separable metric space KK, and the paving K\mathscr{K} of its compact subsets. If π\pi denotes the projection of K×ΩK \times \Omega onto Ω\Omega and

    I(A):=P(π(A)) for all AP(K×Ω),\begin{aligned} I(A) := \mathbb{P}^*(\pi(A)) \text{ for all } A \in \mathfrak{P}(K \times \Omega),\end{aligned}
    then II is a Choquet (KpF)(\mathscr{K} \otimes_p \mathcal{F})-capacity: Monotone increasing property is trivial. To see the ascending property, note that increasing sequence {An}nNP(K×Ω)\{A_n\} _ {n \in \mathbb N} \subseteq \mathfrak{P}(K \times \Omega) we have
    xπ(nAn)      yK such that (y,x)nAn    xnπ(An),\begin{aligned} x \in \pi \left( \bigcup_n A_n \right) \iff \exists \; y \in K \text{ such that } (y,x) \in \bigcup_n A_n \iff x \in \bigcup_n \pi(A_n),\end{aligned}
    showing π(nAn)=nπ(An).\pi \left( \bigcup_n A_n \right) = \bigcup_n \pi(A_n). Thus,
    I(nAn)=P(nπ(An))=supnP(π(An))=supnI(An).\begin{aligned} I\left( \bigcup_n A_n \right) = \mathbb{P}^*\left( \bigcup_n \pi(A_n) \right) = \sup_n \mathbb{P}^*(\pi(A_n)) = \sup_n I(A_n).\end{aligned}
    Finally, for the descending on pavings property, compactness implies that for all ωΩ\omega \in \Omega, (KpF)(ω)(\mathscr{K} \otimes_p \mathcal{F})(\omega) is a compact paving, and thus we can apply Lemma 3 to get
    π(nNEn)=nNπ(En)\begin{aligned} \pi\left(\bigcap _ {n \in \mathbb N} E_n\right) = \bigcap _ {n \in \mathbb N} \pi(E_n)\end{aligned}
    for every decreasing sequence {En}nNKpF.\{E_n\} _ {n \in \mathbb N} \subseteq \mathscr{K} \otimes_p \mathcal{F}. Therefore,
    I(nNEn)=P(nNπ(En)).\begin{aligned} I\left(\bigcap _ {n \in \mathbb N} E_n\right) = \mathbb{P}^* \left(\bigcap _ {n \in \mathbb N} \pi(E_n)\right).\end{aligned}
    Since EnE_n is a finite union of rectangles, π(En)F\pi(E_n) \in \mathcal{F} and therefore,
    P(nNπ(En))=infnP(π(En))=infnI(En).\begin{aligned} \mathbb{P}^* \left(\bigcap _ {n \in \mathbb N} \pi(E_n)\right) = \inf_n \mathbb{P}^* (\pi(E_n)) = \inf_n I(E_n).\end{aligned}

Theorem 6 [Choquet's capacitability theorem]: Consider a paved space (E,E)(E, \mathcal{E}), and let I ⁣:P(E)RI \colon \mathfrak{P}(E) \to \overline{\mathbb R} be a Choquet E\mathcal{E}-capacity. Then every set AE^A \in \widehat{\mathcal{E}} is II-capacitable.

Fix an arbitrary set AE^.A \in \widehat{\mathcal{E}}. If I(A)=I(A) = -\infty then I(K)=I(K) = -\infty for any KAK \subseteq A, and we have our desired equality trivially. Otherwise, we need to show that whenever I(A)>tI(A) > t holds for some given real number tt, there exists a set KEδK \in \mathcal{E}_\delta with KAK \subseteq A and I(K)t.I(K) \ge t.

Recall that

C={BP(E):I(B)>t}\begin{aligned} \mathcal{C} = \{B \in \mathfrak{P}(E)\,:\,I(B) > t\}\end{aligned}
is a capacitance. Then ACA \in \mathcal{C}, and from Sion's Theorem (Theorem 4) there exists a decreasing sequence {Kn}nN\{K_n\} _ {n \in \mathbb N} of elements in EC\mathcal{E} \cap \mathcal{C} such that nNKnA.\bigcap _ {n \in \mathbb N} K_n \subseteq A. But then
I(nNKn)=infnNI(Kn)t,\begin{aligned} I\left(\bigcap _ {n \in \mathbb N} K_n\right) = \inf _ {n \in \mathbb N} I(K_n) \ge t,\end{aligned}
and thus we can take K=nNKn.K = \bigcap_{n \in \mathbb N} K_n.

We are now ready to prove some major results in measure theory.

Measurable Projection

Theorem 7 [Measurable Projection]: Let (Ω,F,P)(\Omega, \mathcal{F}, \mathbb{P}) be a complete probability space, let (K,B(K))(K, \mathcal{B}(K)) be a locally compact separable metric space endowed with the collection of its Borel sets, and denote by π\pi the projection of K×ΩK \times \Omega onto Ω.\Omega. Then, for every BB(K)FB \in \mathcal{B}(K) \otimes \mathcal{F}, the projection π(B)F.\pi(B) \in \mathcal{F}.

We start by noticing B(K)^pF=B(K)F.\mathcal{B}(K) \widehat{\otimes}_p \mathcal{F} = \mathcal{B}(K) \otimes \mathcal{F}. This follows from the fact that if AB(K)pFA \in \mathcal{B}(K) \otimes_p \mathcal{F}, then A=i=1nUi×ViA = \bigcup _ {i=1}^n U_i \times V_i for some UiB(K)U_i \in \mathcal{B}(K) and ViFV_i \in \mathcal{F}, and thus it can be shown AcB(K)^pFA^\mathsf{c} \in \mathcal{B}(K) \widehat{\otimes}_p \mathcal{F}, and thus Lemma 1 gives

B(K)^pF=σ(B(K)pF)=B(K)F.\begin{aligned} \mathcal{B}(K) \widehat{\otimes}_p \mathcal{F} = \sigma(\mathcal{B}(K) \otimes_p \mathcal{F}) = \mathcal{B}(K) \otimes \mathcal{F}.\end{aligned}

Consider the paving K\mathscr{K} on KK consisting of all compact subsets of KK, and introduce the (KpF)(\mathscr{K} \otimes_p \mathcal{F})-capacity I(A)=P(π(A))I(A) = \mathbb{P}^*(\pi(A)), AB(K×Ω)A \in \mathcal{B}(K \times \Omega) we saw before. Now note

B(K)^pF=K^pF,\begin{aligned} \mathcal{B}(K) \widehat{\otimes}_p \mathcal{F} = \mathscr{K} \widehat{\otimes}_p \mathcal{F},\end{aligned}
the mosaic generated by the paving KpF.\mathscr{K} \otimes_p \mathcal{F}. Showing \supseteq is trivial; for \subseteq let AB(K)pFA \in \mathcal{B}(K) \otimes_p \mathcal{F} and show AK^pFA \in \mathscr{K} \widehat{\otimes}_p \mathcal{F} by recalling B(K)=K^.\mathcal{B}(K) = \widehat{\mathscr{K}}.

Choquet's capacitability theorem (Theorem 6) thus guarantees that every set in B(K)F\mathcal{B}(K) \otimes \mathcal{F} is II-capacitable. In particular, for every integer nNn \in \mathbb N, there exists a set Cn(KpF)δC_n \in (\mathscr{K} \otimes_p \mathcal{F})_\delta contained in BB and such that I(Cn)I(B)I(Cn)+(1/n).\begin{aligned} I(C_n) \le I(B) \le I(C_n) + (1/n).\end{aligned} Because Cn(KpF)δC_n \in (\mathscr{K} \otimes_p \mathcal{F})_\delta, CnC_n is a countable intersection Cn=mNGnmC_n = \bigcap _ {m \in \mathbb N} G_n^m, where each GnmG_n^m is a finite union of sets of the form U×VU \times V with UK,VF.U \in \mathscr{K}, V \in \mathcal{F}. Letting Hnm=i=1mGniH_n^m = \bigcap _ {i=1}^m G_n^i, we see that Hn1Hn2H_n^1 \supseteq H_n^2 \supseteq \cdots and Cn=mNHnmC_n = \bigcap _ {m \in \mathbb N} H_n^m, where now HnmH_n^m is also a finite union of sets of the form U×VU \times V with UK,VF.U \in \mathscr{K}, V \in \mathcal{F}.

The form of HnmH_n^m immediately implies π(Hnm)F\pi(H_n^m) \in \mathcal{F} for all (m,n)N2.(m,n) \in \mathbb N^2. Lemma 3 implies

π(Cn)=π(mNHnm)=mNπ(Hnm)F,  nN,\begin{aligned} \pi(C_n) = \pi\left(\bigcap _ {m \in \mathbb N} H_n^m\right) = \bigcap _ {m \in \mathbb N} \pi(H_n^m) \in \mathcal{F}, \quad \forall \; n \in \mathbb N,\end{aligned}
which further implies
π(nNCn)=nNπ(Cn)F.\begin{aligned} \pi\left(\bigcup _ {n \in \mathbb N} C_n\right) = \bigcup _ {n \in \mathbb N} \pi(C_n) \in \mathcal{F}.\end{aligned}
On the other hand, CnBC_n \subseteq B for all nN{n \in \mathbb N} and thus π(nNCn)B.\pi\left(\bigcup _ {n \in \mathbb N} C_n\right) \subseteq B. But (2) implies that the difference of these two sets is a P\mathbb{P}-null set, and the completeness of the probability space gives the desired conclusion π(B)F.\pi(B) \in \mathcal{F}.

It is not easy to construct an example of a Borel set in the product space whose projection is not Borel. It requires study of analytic sets. Check out Corollary 8.2.17 in (Cohn, 2013) for more details.

Measurable Graph

The next theorem is a very visual theorem, especially for the case where KK and Ω\Omega are R.\mathbb R.

Definition 16: For a map f ⁣:XYf \colon X \to Y, we define its graph f\llbracket f \rrbracket to be the product set

f:={(y,x)Y×X:f(x)=y}.\begin{aligned} \llbracket f \rrbracket := \{(y,x) \in Y \times X \,:\, f(x) = y \}.\end{aligned}

We call a set GB(K)FG \in \mathcal{B}(K) \otimes \mathcal{F} a measurable graph, if for every ωΩ\omega \in \Omega its section G(ω)={yK:(y,ω)G}G(\omega) = \{y \in K\,:\,(y,\omega) \in G\} contains at most one point.

Theorem 8 [Measurable Graph]: A subset GG of K×ΩK \times \Omega is a measurable graph, if and only if, there exists a set ΞF\Xi \in \mathcal{F} and a measurable mapping g ⁣:ΞKg \colon \Xi \to K, such that G=g.G = \llbracket g \rrbracket.

Sufficiency: If Ξ\Xi and gg are as stated, the set G={(y,ω)K×Ξ:y=g(ω)}G = \{(y,\omega) \in K \times \Xi\,:\,y = g(\omega)\} equals the pre-image φ1(Δ)\varphi^{-1}(\Delta) of the diagonal Δ={(y,y)K×K:yK}\Delta = \{(y,y) \in K \times K\,:\,y \in K\} under the mapping

K×Ξ(y,ω)φ(y,ω):=(y,g(ω))K×K.\begin{aligned} K \times \Xi \ni (y,\omega) \mapsto \varphi(y,\omega) := (y, g(\omega)) \in K \times K.\end{aligned}
This mapping φ\varphi is (B(K)F)(\mathcal{B}(K) \otimes \mathcal{F})-measurable because of the facts that B(K×K)=B(K)B(K)\mathcal{B}(K \times K) = \mathcal{B}(K) \otimes \mathcal{B}(K), on account of KK being separable, and that gg is measurable. Since Δ\Delta is a closed set (because KK is Hausdorff), ΔB(K×K)\Delta \in \mathcal{B}(K \times K) and thus GG is a measurable graph.

Necessity: Suppose that GG is a measurable graph, and let Ξ:=π(G).\Xi := \pi(G). Then ΞF\Xi \in \mathcal{F} by the Measurable Projection theorem. For every ωΞ\omega \in \Xi, define g(ω)g(\omega) to be the unique element of the set G(ω).G(\omega). We want to show g ⁣:ΞKg \colon \Xi \to K is measurable. Indeed for any HB(K)H \in \mathcal{B}(K), it is easy to see that

g1(H)=π(G(H×Ω))F,\begin{aligned} g^{-1}(H) = \pi\left(G \cap (H \times \Omega)\right) \in \mathcal{F},\end{aligned}
where the inclusion follows from Measurable Projection theorem.

Debut

Now let K=[0,)K = [0, \infty), the case important in stochastic processes.

Definition 17: Let A[0,)×Ω.A \subseteq [0, \infty) \times \Omega. The debut of AA is the nonnegative function DA ⁣:Ω[0,]D_A \colon \Omega \to [0, \infty] defined as
DA(ω)=inf{t[0,):(t,ω)A}.\begin{aligned} D_A(\omega) = \inf\{t \in [0, \infty)\,:\,(t, \omega) \in A\}.\end{aligned}

Recall the convention inf=.\inf \varnothing = \infty. It is easy to see that {DA<}=π(A).\{D_A < \infty\} = \pi(A).

Theorem 9 [Measurable Debut]: Let (Ω,F,P)(\Omega, \mathcal{F}, \mathbb{P}) be a complete probability space, and consider a measurable set AB([0,))F.A \in \mathcal{B}([0, \infty)) \otimes \mathcal{F}. Then the debut DAD_A of this set is a random variable.
For any given real number t>0t > 0, the set DA1([0,t))={ωΩ:DA(ω)<t}D_A^{-1}([0,t)) = \{\omega \in \Omega\,:\,D_A(\omega) < t\} is the projection onto Ω\Omega of the measurable subset A([0,t)×Ω)B([0,))F.A \cap ([0, t) \times \Omega) \in \mathcal{B}([0, \infty)) \otimes \mathcal{F}. To see this, note
ωDA1([0,t))    0DA(ω)<t    s[0,t) such that (s,ω)A    ωπ(A([0,t)×Ω)).\begin{aligned} \omega \in D_A^{-1}([0,t)) \iff 0 \le D_A(\omega) < t \iff \exists \, s \in [0,t) \text{ such that } (s,\omega) \in A \iff \omega \in \pi(A \cap ([0, t) \times \Omega)).\end{aligned}
Measurable Projection theorem shows that this set is in F.\mathcal{F}.

Measurable Section

Let (Ω,F,P)(\Omega, \mathcal{F}, \mathbb{P}) be a complete probability space, and consider a set A[0,)×Ω.A \subseteq [0, \infty) \times \Omega. Then for every ωπ(A),\omega \in \pi(A), there exists a t[0,)t \in [0, \infty) such that (t,ω)A.(t, \omega) \in A. In other words, we can define a mapping Z ⁣:π(A)[0,).Z \colon \pi(A) \to [0, \infty). It is convenient to extend ZZ to the whole of Ω\Omega by setting Z=Z = \infty on Ωπ(A).\Omega \setminus \pi(A). When is it possible to choose ZZ such that it is measurable? The measurable section theorem (also known as measurable selection theorem) says that it is possible to define ZZ to be measurable if AB([0,))F.A \in \mathcal{B}([0, \infty)) \otimes \mathcal{F}.

Recall that for Z ⁣:Ω[0,]Z \colon \Omega \to [0, \infty] its graph is the product set

Z={(t,ω)[0,)×Ω:Z(ω)=t}.\begin{aligned} \llbracket Z \rrbracket = \{(t,\omega) \in [0, \infty) \times \Omega \,:\, Z(\omega) = t\}.\end{aligned}

The condition (Z(ω),ω)A(Z(\omega), \omega) \in A whenever Z<Z < \infty can then be expressed by saying ZA.\llbracket Z \rrbracket \subseteq A.

Other than stochastic processes, measurable section theorems have applications in optimal control and game theory.

Theorem 10 [Measurable Section]: Let (Ω,F,P)(\Omega, \mathcal{F}, \mathbb{P}) be a complete probability space, and consider a measurable set AB([0,))F.A \in \mathcal{B}([0, \infty)) \otimes \mathcal{F}. Then there exists a random variable Z ⁣:Ω[0,]Z \colon \Omega \to [0, \infty] with ZA\llbracket Z \rrbracket \subseteq A and {Z<}=π(A).\{Z < \infty\} = \pi(A).

We divide our analysis into two parts. We shall first show that for every ε>0\varepsilon > 0 there exists a random variable Zε ⁣:Ω[0,]Z_\varepsilon \colon \Omega \to [0, \infty] with ZεA\llbracket Z_\varepsilon \rrbracket \subseteq A and

P(π(A))P(Zε<)+ε.\begin{aligned} \mathbb{P}(\pi(A)) \le \mathbb{P}(Z_\varepsilon < \infty) + \varepsilon.\end{aligned}
To this end, recall that if K=[0,)K = [0, \infty) and K\mathscr{K} is the paving of all compact subsets of KK, then we have the KpF\mathscr{K} \otimes_p \mathcal{F}-capacity I(A)=P(π(A)),AP(K×Ω).I(A) = \mathbb{P}^*(\pi(A)), A \in \mathfrak{P}(K \times \Omega). Therefore, every AB(K)FA \in \mathcal{B}(K) \otimes \mathcal{F} is II-capacitable, and fixing AA, for every ε>0,\varepsilon > 0, there exists a set
Cε(KpF)δ such that CεA,I(Cε)I(A)I(Cε)+ε.\begin{aligned} C_\varepsilon \in (\mathscr{K} \otimes_p \mathcal{F})_\delta \text{ such that } C_\varepsilon \subseteq A, \quad I(C_\varepsilon) \le I(A) \le I(C_\varepsilon) + \varepsilon.\end{aligned}
Let
Zε:=DCε\begin{aligned} Z_\varepsilon := D_{C_\varepsilon}\end{aligned}
be the debut of this set Cε.C_\varepsilon. Then the Measurable Debut theorem (Theorem 9) implies ZεZ_\varepsilon is a random variable.

For every ωΩ\omega \in \Omega, the section Cε(ω)C_\varepsilon(\omega) is a compact subset of KK (use the facts that compact     \iff closed and bounded here, and Cε(KpF)δC_\varepsilon \in (\mathscr{K} \otimes_p \mathcal{F})_\delta). Notice that if (t,ω)DCε(t, \omega) \in \llbracket D_{C_\varepsilon} \rrbracket then DCε(ω)=tD_{C_\varepsilon}(\omega) = t which is same as saying t=inf{s[0,):(s,ω)Cε}t = \inf \{s \in [0, \infty)\,:\,(s,\omega) \in C_\varepsilon\}, but since Cε(ω)C_\varepsilon(\omega) is closed this implies (t,ω)Cε.(t, \omega) \in C_\varepsilon. Therefore, ZεCε\llbracket Z_\varepsilon \rrbracket \subseteq C_\varepsilon, showing the first requirement.

The second requirement P(π(A))P(Zε<)+ε\mathbb{P}\left(\pi(A)\right) \le \mathbb{P}\left(Z_\varepsilon < \infty\right) + \varepsilon is true because π(A)F\pi(A) \in \mathcal{F} by the Measurable Projection theorem (Theorem 7) and {Zε<}=π(Cε)F\{Z_\varepsilon < \infty\} = \pi(C_\varepsilon) \in \mathcal{F}, and now use (2).

Let us now construct a random variable ZZ to satisfy the properties claimed in the theorem. Define

A1:=A.\begin{aligned} A_1 := A.\end{aligned}
Then from above there exists a random variable
Z1 ⁣:Ω[0,] with Z1A1 and P(Z1<)12P(π(A1)),\begin{aligned} Z_1 \colon \Omega \to [0, \infty] \text{ with } \llbracket Z_1 \rrbracket \subseteq A_1 \text{ and } \mathbb{P}(Z_1 < \infty) \ge \frac{1}{2}\mathbb{P}(\pi(A_1)),\end{aligned}
by taking ε=P(Z1<)>0\varepsilon = \mathbb{P}\left(Z_1 < \infty\right) > 0; if it happens that P(Z1<)=0\mathbb{P}\left(Z_1 < \infty\right) = 0, then P(π(A1))=0\mathbb{P}\left(\pi(A_1)\right) = 0 and the inequality is still true. Now define
A2:=A1([0,)×{Z1<}),\begin{aligned} A_2 := A_1 \setminus \left([0, \infty) \times \{Z_1 < \infty\}\right),\end{aligned}
and, reasoning as before, construct a random variable
Z2 ⁣:Ω[0,] with Z2A2 and P(Z2<)12P(π(A2))=12(P(π(A1))P(Z1<)).\begin{aligned} Z_2 \colon \Omega \to [0, \infty] \text{ with } \llbracket Z_2 \rrbracket \subseteq A_2 \text{ and } \mathbb{P}(Z_2 < \infty) &\ge \frac{1}{2}\mathbb{P}(\pi(A_2)) \\ &= \frac{1}{2} \left( \mathbb{P}(\pi(A_1)) - \mathbb{P}(Z_1 < \infty) \right).\end{aligned}
Continuing this way, we obtain a sequence {Zn}nN\{Z_n\} _ {n \in \mathbb N} such that ZnA\llbracket Z_n \rrbracket \subseteq A, the projections π(Zn)\pi(\llbracket Z_n \rrbracket) are disjoint, and we have k=1nP(Zk<)(12+122++12n)P(π(A1))=(12n)P(π(A)),nN.\begin{aligned} \sum_{k=1}^n \mathbb{P}(Z_k < \infty) \ge \left( \frac{1}{2} + \frac{1}{2^2} + \cdots + \frac{1}{2^n} \right) \mathbb{P}(\pi(A_1)) = (1 - 2^{-n})\mathbb{P}(\pi(A)), \quad \forall n \in \mathbb N.\end{aligned} Therefore, the random variable ZZ defined as
Z(ω)={Zk(ω)if ω{Zk<} for some kNotherwise,\begin{aligned} Z(\omega) = \begin{cases} Z_k(\omega) &\text{if } \omega \in \{Z_k < \infty\} \text{ for some } k \in \mathbb N \\ \infty &\text{otherwise,} \end{cases}\end{aligned}
satisfies ZA\llbracket Z \rrbracket \subseteq A, thus also {Z<}π(A).\{Z < \infty\} \subseteq \pi(A).

On the other hand, letting nn \to \infty in (3), we see that {Z<}\{Z < \infty\} and π(A)\pi(A) have the same probability. Therefore, the completeness of the probability space implies these two sets can be made equal.

Epilogue

With this we are done laying the foundations. In the next blog post, we will discuss applications of these results in the general theory of processes.

References

[1] If you steal from one author, it’s plagiarism; if you steal from many, it’s research.