Borel measures on polar sets

We define the cofine topology in the present setting, which shows that it satisfies the conditions introduced by J. B. Walsh. We modify such a topology on a left polar set and it must have a cofine topology. The definition introduced here is canonical. We combine the exceptional sets N, N_1, and N into one set, which we denote it by N.

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Introduction

The terminology Borel measure is used by different authors with different meanings: (A) Some authors use it for measures 𝜇 on the σ-algebra 𝐵 of Borel subsets of a given topological space 𝑋, i.e. functions 𝜇: 𝐵 → [0, ∞] which are countably additive.

(B) Some authors use it for measures 𝜇 on the 𝜎-algebra of Borel sets of a locally compact topological space satisfying the additional property 𝑡ℎ𝑎𝑡 𝜇(𝐾) < ∞ for every compact set 𝐾.

(C) Some authors use it for outer measures 𝜇 on a topological space 𝑋 for which the Borel sets are 𝜇-measurable (hence the difference between acception (A) and (B) is small, however when coming to the terminology Borel regular we will see more important discrepancies).

Borel Regular Measures

In these three different contexts Borel regular measures are then defined as follows: (A) Borel measures 𝜇 for which 𝑠𝑢𝑝{𝜇(𝐶): 𝐶 ⊂ 𝐸 𝑖𝑠 𝑐𝑙𝑜𝑠𝑒𝑑} = 𝜇(𝐸) any Borel set 𝐸. 𝑠𝑢𝑝{𝜇(𝐶): 𝐶 ⊂ 𝐸 𝑖𝑠 𝑐𝑙𝑜𝑠𝑒𝑑} = 𝜇(𝐸) for any Borel set 𝐸. Terminology: Using terminology (A) call tight the measures called Borel regular by authors using terminology (B). Moreover they call τ-smooth those Borel measures for which 𝜇(𝐹𝑘) → 0𝜇(𝐹𝑘) → 0 for any sequence (or, more in general, net) of closed sets with 𝐹𝑘 ↓ ∅𝐹𝑘 ↓ ∅.

The study of Borel measures is often connected with that of Baire measures, which differ from Borel measures only in their domain of definition: they are defined on the smallest σσ-algebra 𝛽 0 for which continuous functions are 𝛽 0 measurable. In particular observe that the two concepts coincide on topological spaces such that for any open set 𝑈 there is a continuous function 𝑓 with 𝑓

The concept of tightness and τ-smoothness can be extended to Baire measures as well (and in fact using terminology (B) call Baire regular the Baire measures which are tight; If 𝑋 is a completely regular space, then any τ-smooth (tight) finite Baire measure can be extended to a regular τ-smooth (tight) finite Borel measure

Polar Set

The polar set of analytic function 𝑓(𝑧) of the complex variable 𝑧 = (𝑧 1 , … , 𝑧 𝑛 ), 𝑛 ≥ 1 is the set 𝑃 of points in some domain 𝐷 of the complex space 𝐶 𝑛 for which: a) 𝑓(𝑧) is holomorphic everywhere in 𝐷 ∖ 𝑃; b) 𝑓(𝑧) cannot be analytically continued to any point of 𝑃; and c) for every point Any polar set is totally thin but, in contrast with classical potential theory, not every totally thin set is polar. A similar theory of polarity holds in harmonic spaces, or in the more general case of balayage spaces. In probabilistic potential theory, a Borel set is polar if its first hitting time 𝑇 𝐴 satisfies 𝑇 𝐴 = ∞ a.s.

Definition

A Borel set 𝐴 is elusive at a points 𝑦 ∈ 𝐸 -𝑁 if 𝑦𝑃 𝐴 (𝑥) < 1 for some 𝑥 ∈ 𝐸 𝑔 . We say 𝐴 is elusive at a points 𝑦 ∈ 𝑁 if 𝑦 is not in the ordinary closure of 𝐴. The definition of elusiveness at points of the exceptional set 𝑁 may seem arbitrary, but 𝑁 is left polar. We observe that if 𝑦 ∈ 𝐴 -𝑁 then 𝐴 is not elusive at y (for 𝑦 𝑃 { 𝜏 < ∞, 𝑋 𝜏 = 𝑦 ∈ 𝐴 } = 1 for all 𝑥 ∈ 𝐸 g ). Hence, we determine that the idea of elusiveness is of interest only when 𝑦 ∉ 𝐴.

Proposition

Let 𝑦 ∈ 𝐸 -𝑁 and let 𝐴 ∈ 𝛽 such that 𝑦 ∈ 𝐴. Then 𝐴 is elusive at

Now {𝐿 𝐴 < 𝜏 } is invariant, so the right-hand side of ( 2) is either zero or one.

If 𝐴 is elusive, the left-hand side is strictly positive for some 𝑥, which implies that (1) holds. Conversely if (1) holds, there exists 𝑡 such that

Thus y𝑃 𝐴 ∉ 1, and 𝐴 is elusive at 𝑦.

Hence the assumption is valid. Hence, if 𝐴 is elusive at 𝑦, then 𝑋 𝑡 -is in 𝐸 -𝐴 for all 𝑡 sufficiently close to 𝜌.

Definition

A Borel set 𝑉 ⊂ 𝐸 is a cofine neighborhood of 𝑦 ∈ 𝐸 is 𝑦 ∈ 𝑉 and if 𝐸 -𝑉 is elusive at 𝑦. It is evident from (1) that the intersection of two cofine neighborhoods is a cofine neighborhood. Thus the cofine neighborhood from a neighborhood base for a topology on 𝐸, called the cofine topology. We extend this to 𝐸 ∪ 𝛿, by making 𝛿 an isolated point.

We find that any ordinary neighborhood of 𝑦 is a cofine neighborhood. Thus the cofine topology is finer than the ordinary topology.

Theorem

Let 𝐴 be Borel. Then the cofine closure of 𝐴 is also Borel.

Proof

Let 𝜇o be a reference measure. Let 𝐿 be an 0-measurable time which is y𝑃 𝜇 0 -a.s. equal to 𝐿 𝐴 , for each 𝑦. In view of assumptions of (2.) cofine closure of 𝐴 is

This is Borel structure. We now prove that the name "cofine topology" is justified.

Theorem

The cofine topology defined in ( 2) is a cofine topology, i.e., it satisfies conditions of (1).

Proof

Let 𝐿 be a bounded Borel function which is cofine continuous, except possibly at a left polar set. We show 𝑡 → f (𝑋 𝑡 ) is left continuous for it is sufficient to show it is left continuous on (0, 𝐿 𝐾 ),where 𝐾 is any compact set. Let 𝑋 be the right continuous process { 𝑋 ̅ t = 𝑋 𝐿 𝐾-𝑡 -, 0 ≤ 𝑡 ≤ 𝐿 𝐾 }. It is then sufficient to show f (𝑋 𝑡 ) is right continuous, and even sufficient to show f (𝑋 𝑡 ) is right continuous at a given stopping time 𝑇. But (𝐿 𝐾 -𝑇) ∨ 0 is co-optional for the original process. Hence, it is sufficient to show that for any co-optional time 𝐿,

If 𝐿 is a finite co-optional time, the process killed at 𝐿 is an ℎ -transform, with ℎ(𝑥) = 𝑃 𝑥 (𝐿 > 0). Since 𝑋 𝐿 -exists, implies that h is a Green's potential, say ℎ = 𝐺 𝜇 . We find that,

where the penultimate equality follows from the equation (5 We combine equations ( 7) and ( 8) which equal =

But now 𝑠 → f (𝑋 𝑠 -) has left limits 𝑃 𝑥a. s., so 𝑃 𝑥 { 0 < 𝐿 𝐾 < ∞ } lim ̅̅̅̅ 𝑠↑𝜌 𝑓 (𝑋 𝑠 -) > lim 𝑠↑𝜌 𝑓 (𝑋 𝑠 -). This contradicts (7) unless 𝐸 𝑃 𝐾 = Ø, i.e., unless 𝐾 is left polar. 𝐾 is arbitrary, so it follows that 𝐴 is left polar. Thus the set at which f fails to have a cofine limit is left polar. Now the set {: 𝑓(𝑥) ̅̅̅̅̅̅ < 𝑓 (𝑥) -𝜀} can have no cofine accumulation points, from which it follows that f is cofine contradicts except at A. By (CFI), which we have verified, both 𝑠f (𝑋 𝑠 -) and 𝑠 -𝑋 𝑠 -are left continuous.

We consider the two left continuous processes 𝑌 𝑡 = lim But we know that lim 𝑠↑𝜌 𝑓(𝑋 𝑠 -) exists, so if y is not in a left polar set, 𝑦 𝑃 𝑋 { ∧ } = 1, x ∈ E g , ; hence the last term in ( 9) is one.

Hence, the theorem is proved.

There are different topologies associated with the topology on sets and Getoor's 𝑑 -topology, and extend some of the results.

We define a topology as follows: Let us consider a Borel set 𝑉 is a neighborhood of 𝑦 ∈ 𝐸 -𝑁 in the 𝜏 -topology if 𝑦 ∈ 𝑉 and if 𝑉 𝑐 is co-thin at 𝑦, and 𝑉 is a neighborhood of 𝑦 ∈ 𝑁 if it is an ordinary neighborhood of 𝑦. These neighborhoods from a base for a topology which we denote it by 𝜏 -topology.

We compare the 𝜏 -topology with the cofine topology, with the observations that it 𝑀 𝐴 and 𝐿 𝐴 are respectively the last exits of 𝑋 𝑡 and X 𝑡 -from 𝐴, then 𝑉 is an 𝜏 -neighborhood of 𝑦 iff 𝑦 ∈ 𝑉 and y𝑃 𝑋 { 𝑀 𝑉 𝐶 < 𝜏, 𝜏 > 0 }, = 0 ……11.

while 𝑉 is a cofine neighborhood iff 𝑦 ∈ 𝑉 and y𝑃 𝑋 { 𝐿 𝑉 𝐶 < 𝜌, 𝜌 > 0 } = 0. …….12.

The difference between the two only involves interchanging 𝑋 𝑡 and 𝑋 𝑡 -but it is interesting feature that 𝜏 -topology is not always a cofine topology. However, the two topologies coincide in the case 𝑋 is continuous and in fact they coincide in the case of standard processes in duality, as we will see.

We know that 𝑋 is said to satisfy Runt's hypothesis (B) if, for each semi polar 𝐵, { 𝑃 𝑥 ∃t: 𝑋 𝑡 ∈ 𝐵 and 𝑋 𝑡 -≠ 𝑋 𝑡 = 0, 𝑎𝑙𝑙 𝑥 ∈ 𝐸}.

Theorem

Suppose that 𝑋 is quasi-left continuous, and satisfies Runt's hypothesis. Then the 𝜏 -topology and the cofine topology coincide. Before proving the theorem (6), we consider another theorem whose assertions are suitably applied. We generalize here the results due to J. B. Walsh.

Theorem

Theorem

Proof

We consider the case where A is compact. Suppose 𝑋 is quasi-left continuous and left f (𝑥) = 𝐸 𝑋 {𝑒 -𝑆 𝐴 }. 𝑆 𝐴 is predictable on 𝑋 𝑆 𝐴 ∉ 𝐴, hence 𝑋 𝑆 𝐴 = 𝑋 𝑆 𝐴 ∈ 𝐴, while on the set 𝑋 𝑆 𝐴 ∈ 𝐴, we must have f (𝑋 𝑆 𝐴 ) = 1 by the strong Markov property. But {𝑥: 𝑓 (𝑥) = 1} ⊂ 𝐴. Thus in either case 𝑇 𝐴 ≤ 𝑆 𝐴 .

Suppose 𝑋 satisfies Runt's Hypothesis and let 𝑈 𝐴 be the terminal time

And let ℎ(𝑥) = 𝐸 𝑋 { 𝑒 -𝑈 𝐴 }, and let 𝐴′ = { 𝑥: ℎ(𝑥) = 1}. Suppose we further show that 𝐴 -𝐴′ is semi polar. Since 𝐴 is compact, 𝑋 𝑇 𝐴 ∈ A′, then, since 𝑆 𝐴 ≤ 𝑈 𝐴 , 𝑆 𝐴 °𝜃𝑇 = 0 a.s., so 𝑆 𝐴 ≤ 𝑇 𝐴 . On the other hand, 𝑃 𝑋 { 𝑋 𝑇 ∈ A -A′, 𝑋 𝑇 ≠ 𝑋 𝑇 } = 0, and again 𝑆 𝐴 ≤ 𝑇 𝐴 . We further show that 𝐴 -𝐴′ is semi polar. Consider the set 𝐵 = 𝐴 ∩ { 𝑥: 𝑓 (𝑥) < 1 -𝜀}, and let 𝑈 1 = 𝑈 𝐵 , 𝑈 𝑛+1 = 𝑈 𝑛 + 𝑈 𝐵 o 𝜃 𝑈 𝑛 , 𝐴 is compact and 𝐵 is fine closed, hence 𝑋 𝑈 𝑛 ∈ 𝐵 if 𝑈 𝑛 < ∞. Since 𝐵 ⊆ 𝐴, 𝑈 𝐵 ≥ 𝑈 𝐴 , so on { 𝑈 𝑛 < ∞ }, 𝐸{𝑒 -(𝑈 𝑛+1 -𝑈 𝑛 ) / } ≤ 1 -𝜀.𝜏 𝑈 𝑛 } ≤ 1 -𝜀 It thus follows that 1im 𝑈 𝑛 = ∞. But we consider the interval (𝑈 𝑛 , 𝑈 𝑛+1 ). There cannot be at in this interval for which 𝑋 𝑡 ∈ 𝐵 and 𝑋 𝑡 -= 𝑋 𝑡 . Since 𝑋 𝑡 is continuous except on a countable set, 𝑋 𝑡 can be in 𝐵 for at most countably many t ∈ (𝑈 𝑛 , 𝑈 𝑛+1 ), and hence for at most contably many 𝑡 ≥ 0. According to Dellacherie 𝐵 is semipolar. Then so is 𝐴 -𝐴′, which is a countable union of such sets. Hence, the theorem is proved. Thus conditions (11) and ( 12) reduce to the same condition. Hence, the theorem is proved. An important topology, called the d-topology, was introduced by Getoor in order to study regular excessive functions. We examine that it also coincide with the cofine topology. The d-open set is a cofine neighborhood of quasi every one of its points. Since 𝐴 𝑐 is elusive, hence co-thin, at all 𝑦 ∈ 𝐾.

Proof of

Proof of

Definition

We suppose 𝐴 𝑐 is 𝑑 -set and let 𝐾 be a compact subset of those 𝑦 ∈ 𝐴 for which 𝐴 𝑐 is not elusive (and hence not co-thin). The process killed at 𝐿 𝐾 is a 𝑃 𝐾 -transform, and 𝑃 𝐾 = 𝐺 𝜇 with supp (µ) ⊂ 𝐾, so 𝑃 𝑥 { 𝑀 𝐴 𝐶 (𝐿 𝐾 ) = 𝐿 𝐾 > 0 } = ∫K g 𝑦 (𝑥) y𝑃 𝑋 { 𝑀 𝐴 𝐶 = 𝜌 } µ(𝑑𝑦) = ∫K g 𝑦 (x) µ(dy) = 𝑃 𝐾 (x), …….16.

Since 𝐴 is not a co-fine neighborhood of any 𝑦 ∈ 𝐾. Thus if 𝐿 𝐾 > 0, then 𝑋 𝐿 𝐾 ∈ 𝐾 and 𝑀 𝐴 𝐶 (𝐿 𝐾 ) = 𝐿 𝐾 i.e., [L] ⊂ ∧ 𝐴 𝑐 , K. Therefore ∧ 𝐴 𝑐 , K is not evanescent unless 𝐾 is left polar.

It is interesting feature that the d-topology is strictly finer then the co-fine topology at every point, even in the best of cases. We illustrate it by considering Brownian motion in the plane. If 𝑥 is any point and 𝐷 a line passing through 𝑥, 𝐷, being closed, is a dset. Since points are polar, 𝐷′ = 𝐷 -{𝑥} is still a 𝑑 -set. Thus 𝑅 2 -𝐷′ is a neighborhood of x in the d-topology, but it is not a co-fine neighborhood. Hence, the fine and co-fine topology coincide in this case, and it is known that Brownian motion from x will encounter 𝐷′.

We discuss here elementary properties of Green's function in the context of results derived by 𝐾. Janssen on the existence of harmonic space.

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