Part 1 Classical and Parabolic Potential Theory

Chapter Ⅰ Introduction to the Mathematical Background of Classical Potential Theory

1．The Context of Green's Identity

2．Function Averages

3．Harmonic Functions

4．Maximum-Minimum Theorem for Harmonic Functions

5．The Fundamental Kernel for RN and Its Potentials

6．Gauss Integral Theorem

7．The Smoothness of Potentials;The Poisson Equation

8．Harmonic Measure and the Riesz Decomposition

Chapter Ⅱ Basic Properties of Harmonic,Subharmonic,and Superharmonic Functions

1．The Green Function of a Ball;The Poisson Integral

2．Harnack's Inequality

3．Convergence of Directed Sets of Harmonic Functions

4．Harmonic,Subharmonic,and Superharmonic Functions

5．Minimum Theorem for Superharmonic Functions

6．Application of the Operation τB

7．Characterization of Superharmonic Functions in Terms of Harmonic Functions

8．Differentiable Superharmonic Functions

9．Application of Jensen's Inequality

10．Superharmonic Functions on an Annulus

11．Examples

12．The Kelvin Transformation(N≥2)

13．Greenian Sets

14 The L1(μB-)and D(μB-)Classes of Harmonic Functions on a Ball B;The Riesz-Herglotz Theorem

15．The Fatou Boundary Limit Theorem

16．Minimal Harmonic Functions

Chapter Ⅲ Infima of Families of Superharmonic Functions

1．Least Superharmonic Majorant(LM) and Greatest Subharmonic Minorant(GM)

2．Generalization of Theorem 1

3．Fundamental Convergence Theorem(Preliminary Version)

4．The Reduction Operation

5．Reduction Properties

6．A Smallness Property of Reductions on Compact Sets

7．The Natural(Pointwise)Order Decomposition for Positive Superharmonic Functions

Chapter Ⅳ Potentials on Special Open Sets

1．Special Open Sets,and Potentials on Them

2．Examples

3．A Fundamental Smallness Property of Potentials

4．Increasing Sequences of Potentials

5．Smoothing of a Potential

6．Uniqueness of the Measure Determining a Potential

7．Riesz Measure Associated with a Superharmonic Function

8．Riesz Decomposition Theorem

9．Counterpart for Superharmonic Functions on R2 of the Riesz Decomposition

10 An Approximation Theorem

Chapter Ⅴ Polar Sets and Their Applications

1．Definition

2．Superharmonic Functions Associated with a Polar Set

3．Countable Unions of Polar Sets

4．Properties of Polar Sets

5．Extension of a Superharmonic Function

6．Greenian Sets in R2 as the Complements of Nonpolar Sets

7．Superharmonic Function Minimum Theorem(Extension of Theorem II.5)

8．Evans-Vasilesco Theorem

9．Approximation of a Potential by Continuous Potentials

10．The Domination Principle

11．The Infinity Set of a Potential and the Riesz Measure

Chapter Ⅵ The Fundamental Convergence Theorem and the Reduction Operation

1．The Fundamental Convergence Theorem

2．Inner Polar versus Polar Sets

3．Properties of the Reduction Operation

4．Proofs of the Reduction Properties

5．Reductions and Capacities

Chapter Ⅶ Green Functions

1．Definition of the Green Function GD

2．Extremal Property of GD

3．Boundedness Properties of GD

4．Further Properties of GD

5．The Potential GDμ of a Measure μ

6．Increasing Sequences of Open Sets and the Corresponding Green Function Sequences

7．The Existence of GD versus the Greenian Character of D

8．From Special to Greenian Sets

9．Approximation Lemma

10．The Function GD(·,ζ)|D-{ζ} as a Minimal Harmonic Function

Chapter Ⅷ The Dirichlet Problem for Relative Harmonic Functions

1．Relative Harmonic,Superharmonic,and Subharmonic Functions

2．The PWB Method

3．Examples

4．Continuous Boundary Functions on the Euclidean Boundary(h≡1)

5．h-Harmonic Measure Null Sets

6．Properties of PWBh Solutions

7．Proofs for Section 6

8．h-Harmonic Measure

9．h-Resolutive Boundaries

10．Relations between Reductions and Dirichlet Solutions

11．Generalization of the Operator τh B and Application to GMh

12．Barriers

13．h-Barriers and Boundary Point h-Regularity

14．Barriers and Euclidean Boundary Point Regularity

15．The Geometrical Significance of Regularity(Euclidean Boundary,h≡1)

16．Continuation of Section 13

17．h-Harmonic Measure μh D as a Function ofD

18．The Extension G＝ D of GD and the Harmonic Average μD(ξ,G＝ B(η,·))When D ？ B

19．Modification of Section 18 for D=R2

20．Interpretation of φD as a Green Function with Pole ∞(N=2)

21．Variant of the Operator τB

Chapter Ⅸ Lattices and Related Classes of Functions

1．Introduction

2．LMh D u for an h-Subharmonic Function u

3．The Class D(μh D-)

4．The Class Lp(μh D-)(p≥1)

5．The Lattices(S±,≤)and(S+,≤)

6．The Vector Lattice (S,≤)

7．The Vector Lattice Sm

8．The Vector Lattice Sp

9．The Vector Lattice Sqb

10．The Vector Lattice Ss

11．A Refinement ofthe Riesz Decomposition

12．Lattices of h-Harmonic Functions on a Ball

Chapter Ⅹ The Sweeping Operation

1．Sweeping Context and Terminology

2． Relation between Harmonic Measure and the Sweeping Kernel

3．Sweeping Symmetry Theorem

4．Kernel Property of δA D

5．Swept Measures and Functions

6．Some Properties of δA D

7．Poles of a Positive Harmonic Function

8．Relative Harmonic Measure on a Polar Set

Chapter Ⅺ The Fine Topology

1．Definitions and Basic Properties

2．A Thinness Criterion

3．Conditions That ξ∈A∫

4．An Internal Limit Theorem

5．Extension of the Fine Topology to RN∪{∞}

6．The Fine Topology Derived Set of a Subset of RN

7．Application to the Fundamental Convergence Theorem and to Reductions

8．Fine Topology Limits and Euclidean Topology Limits

9．Fine Topology Limits and Euclidean Topology Limits(Continued)

10．Identification of A∫ in Terms ofa Special Functionu

11．Quasi-Lindel？f Property

12．Regularity in Terms of the Fine Topology

13．The Euclidean Boundary Set of Thinness of a Greenian Set

14．The Support of a Swept Measure

15．Characterization of ‖μ‖A

16．A Special Reduction

17．The Fine Interior of a Set of Constancy of a Superharmonic Function

18．The Support of a Swept Measure (Continuation of Section 14)

19．Superharmonic Functions on Fine-Open Sets

20．A Generalized Reduction

21．Limits of Superharmonic Functions at Irregular Boundary Points of Their Domains

22．The Limit Harmonic Measure ∫μD

23．Extension of the Domination Principle

Chapter Ⅻ The Martin Boundary

1．Motivation

2．The Martin Functions

3．The Martin Space

4．Preliminary Representations of Positive Harmonic Functions and Their Reductions

5．Minimal Harmonic Functions and Their Poles

6．Extension of Lemma 4

7．The Set of Nonminimal Martin Boundary Points

8．Reductions on the Set of Minimal Martin Boundary Points

9．The Martin Representation

10．Resolutivity of the Martin Boundary

11．Minimal Thinness at a Martin Boundary Point

12．The Minimal-Fine Topology

13．First Martin Boundary Counterpart of Theorem XI.4(c)and(d)

14．Second Martin Boundary Counterpart of Theorem XI.4(c)

15．Minimal-Fine Topology Limits and Martin Topology Limits at a Minimal Martin Boundary Point

16．Minimal-Fine Topology Limits and Martin Topology Limits at a Minimal Martin Boundary Point(Continued)

17．Minimal-Fine Martin Boundary Limit Functions

18．The Fine Boundary Function of a Potential

19．The Fatou Boundary Limit Theorem for the Martin Space

20．Classical versus Minimal-Fine Topology Boundary Limit Theorems for Relative Superharmonic Functions on a Ball in RN

21．Nontangential and Minimal-Fine Limits at a Half-space Boundary

22．Normal Boundary Limits for a Half-space

23．Boundary Limit Function (Minimal-Fine and Normal)of a Potential on a Half-space

Chapter ⅩⅢ Classical Energy and Capacity

1．Physical Context

2．Measures and Their Energies

3．Charges and Their Energies

4．Inequalities between Potentials,and the Corresponding Energy Inequalities

5．The Function D？GDμ

6．Classical Evaluation of Energy;Hilbert Space Methods

7．The Energy Functional(Relative to an Arbitrary Greenian Subset D of RN)

8．Alternative Proofs ofTheorem 7(b+)

9．Sharpening of Lemma 4

10．The Classical Capacity Function

11．Inner and Outer Capacities(Notation of Section 10)

12．Extremal Property Characterizations of Equilibrium Potentials(Notation of Section 10)

13．Expressions for C(A)

14．The Gauss Minimum Problems and Their Relation to Reductions

15．Dependence of C on D

16．Energy Relative to R2

17．The Wiener Thinness Criterion

18．The Robin Constant and Equilibrium Measures Relative to R2(N=2)

Chapter ⅩⅣ One-Dimensional Potential Theory

1．Introduction

2．Harmonic,Superharmonic,and Subharmonic Functions

3．Convergence Theorems

4．Smoothness Properties of Superharmonic and Subharmonic Functions

5．The Dirichlet Problem(Euclidean Boundary)

6．Green Functions

7．Potentials of Measures

8．Identification of the Measure Defining a Potential

9．Riesz Decomposition

10．The Martin Boundary

Chapter ⅩⅤ Parabolic Potential Theory:Basic Facts

1．Conventions

2．The Parabolic and Coparabolic Operators

3．Coparabolic Polynomials

4．The Parabolic Green Function of RN

5．Maximum-Minimum Parabolic Function Theorem

6．Application of Green's Theorem

7．The Parabolic Green Function of a Smooth Domain;The Riesz Decomposition and Parabolic Measure(Formal Treatment)

8．The Green Function of an Interval

9．Parabolic Measure for an Interval

10．Parabolic Averages

11．Harnack's Theorems in the Parabolic Context

12．Superparabolic Functions

13．Superparabolic Function Minimum Theorem

14．The Operation ？ and the Defining Average Properties of Superparabolic Functions

15．Superparabolic and Parabolic Functions on a Cylinder

16．The Appell Transformation

17．Extensions of a Parabolic Function Defined on a Cylinder

Chapter ⅩⅥ Subparabolic,Superparabolic,and Parabolic Functions on a Slab

1．The Parabolic Poisson Integral fora Slab

2．A Generalized Superparabolic Function Inequality

3．A Crrterion of a Subparabolic Function Supremum

4．A Boundary Limit Criterion for the Identically Vanishing of a Positive Parabolic Function

5．A Condition that a Positive Parabolic Function Be Representable by a Poisson Integral

6．The L1(？-)and D(？-)Classes of Parabolic Functions on a Slab

7．The Parabolic Boundary Limit Theorem

8．Minimal Parabolic Functions on a Slab

Chapter ⅩⅦ Parabolic Potential Theory (Continued)

1．Greatest Minorants and Least Majorants

2．The Parabolic Fundamental Convergence Theorem(Preliminary Version)and the Reduction Operation

3．The Parabolic Context Reduction Operations

4．The Parabolic Green Function

5．Potentials

6．The Smoothness of Potentials

7．Riesz Decomposition Theorem

8．Parabolic-Polar Sets

9．The Parabolic-Fine Topology

10．Semipolar Sets

11．Preliminary List of Reduction Properties

12．A Criterion of Parabolic Thinness

13．The Parabolic Fundamental Convergence Theorem

14．Applications of the Fundamental Convergence Theorem to Reductions and to Green Functions

15．Applications of the Fundamental Convergence Theorem to the Parabolic-Fine Topology

16．Parabolic-Reduction Properties

17．Proofs of the Reduction Properties in Section 16

18．The Classical Context Green Function in Terms of the Parabolic Context Green Function(N≥1)

19．The Quasi-Lindel？f Property

Chapter ⅩⅧ The Parabolic Dirichlet Problem,Sweeping,and Exceptional Sets

1．Relativization of the Parabolic Context;The PWB Method in this Context

2．h-Parabolic Measure

3．Parabolic Barriers

4．Relations between the Classical Dirichlet Problem and the Parabolic Context Diriehlet Problem

5．Classical Reductions in the Parabolic Context

6．Parabolic Regularity of Boundary Points

7．Parabolic Regularity in Terms of the Fine Topology

8．Sweeping in the Parabolic Context

9．The Extension ？of ？ and the Parabolic Average ？(？,？(·,？)when ？

10．Conditions that ξ∈？ps

11．Parabolic-and Coparabolic-Polar Sets

12．Parabolic-and Coparabolic-Semipolar Sets

13．The Support of a Swept Measure

14．An Internal Limit Theorem;The Coparabolic-Fine Topology Smoothness of Superparabolic Functions

15．Application to a Version of the Parabolic Context Fatou Boundary Limit Theorem on a Slab

16．The Parabolic Context Domination Principle

17．Limits of Superparabolic Functions at Parabolic-Irregular Boundary Points of Their Domains

18．Martin Flat Point Set Pairs

19．Lattices and Related Classes of Functions in the Parabolic Context

Chapter ⅩⅨ The Martin Boundary in the Parabolic Context

1．Introduction

2．The Martin Functions of Martin Point Set and Measure Set Pairs

3．The Martin Space ？M

4．Preparatory Material for the Parabolic Context Martin Representation Theorem

5．Minimal Parabolic Functions and Their Poles

6．The Set of Nonminimal Martin Boundary Points

7．The Martin Representation in the Parabolic Context

8 Martin Boundary of a Slab ？=RN×]0,δ[with 0＜δ≤+∞

9．Martin Boundaries for the Lower Half-space of？Nand for ？N

10．The Martin Boundary of？=]0,+∞[×]-∞,δ[

11．？WB？Solutions on ？M

12．The Minimal-Fine Topology in the Parabolic Context

13．Boundary Counterpart ofTheorem ⅩⅧ.14(f)

14．The Vanishing ofPotentials on ？M？

15．The Parabolic Context Fatou Boundary Limit Theorem on Martin Spaces

Part 2 Probabilistic Counterpart of Part1

Chapter Ⅰ Fundamental Concepts of Probability

1．Adapted Families of Functions on Measurable Spaces

2．Progressive Measurability

3．Random Variables

4．Conditional Expectations

5．Conditional Expectation Continuity Theorem

6．Fatou's Lemma for Conditional Expectations

7．Dominated Convergence Theorem for Conditional Expectations

8．Stochastic Processes,“Evanescent,""Indistinguishable,""Standard Modification,""Nearly"

9．The Hitting of Sets and Progressive Measurability

10．Canonical Processes and Finite-Dimensional Distributions

11．Choice of the Basic Probability Space

12．The Hitting of Sets by a Right Continuous Process

13．Measurability versus Progressive Measurability of Stochastic Processes

14．Predictable Families of Functions

Chapter Ⅱ Optional Times and Associated Concepts

1．The Context of Optional Times

2．Optional Time Properties(Continuous Parameter Context)

3．Process Functions at Optional Times

4．Hitting and Entry Times

5．Application to Continuity Properties of Sample Functions

6．Continuation of Section 5

7．Predictable Optional Times

8．Section Theorems

9．The Graph of a Predictable Time and the Entry Time of a Predictable Set

10．Semipolar Subsets of R+×Ω

11．The Classes D and Lp of Stochastic Processes

12．Decomposition of Optional Times;Accessible and Totally Inaccessible Optional Times

Chapter Ⅲ Elements of Martingale Theory

1．Definitions

2．Examples

3．Elementary Properties(Arbitrary Simply Ordered Parameter Set)

4．The Parameter Set in Martingale Theory

5．Convergence of Supermartingale Families

6．Optional Sampling Theorem(Bounded Optional Times)

7．Optional Sampling Theorem for Right Closed Processes

8．Optional Stopping

9．Maximal Inequalities

10．Conditional Maximal Inequalities

11．An Lp Inequality for Submartingale Suprema

12．Crossings

13．Forward Convergence in the L1 Bounded Case

14．Convergence ofa Uniformly Integrable Martingale

15．Forward Convergence of a Right Closable Supermartingale

16．Backward Convergence of a Martingale

17．Backward Convergence of a Supermartingale

18．The τ Operator

19．The Natural Order Decomposition Theorem for Supermartingales

20．The Operators LM and GM

21．Supermartingale Potentials and the Riesz Decomposition

22．Potential Theory Reductions in a Discrete Parameter Probability Context

23．Application to the Crossing Inequalities

Chapter Ⅳ Basic Properties of Continuous Parameter Supermartingales

1．Continuity Properties

2．Optional Sampling of Uniformly Integrable Continuous Parameter Martingales

3．Optional Sampling and Convergence of Continuous Parameter Supermartingales

4．Increasing Sequences of Supermartingales

5．Probability Version of the Fundamental Convergence Theorem of Potential Theory

6．Quasi-Bounded Positive Supermartingales;Generation of Supermartingale Potentials by Increasing Processes

7．Natural versus Predictable Increasing Processes(I=Z+ or R+)

8．Generation of Supermartingale Potentials by Increasing Processes in the Discrete Parameter Case

9．An Inequality for Predictable Increasing Processes

10．Generation of Supermartingale Potentials by Increasing Processes for Arbitrary Parameter Sets

11．Generation of Supermartingale Potentials by Increasing Processes in the Continuous Parameter Case:The Meyer Decomposition

12．Meyer Decomposition of a Submartingale

13．Role of the Measure Associated with a Supermartingale;The Supermartingale Domination Principle

14．The Operators τ,LM,and GM in the Continuous Parameter Context

15．Potential Theory on R+×Ω

16．The FineTopology of R+× Ω

17．Potential Theory Reductions in a Continuous Parameter Probability Context

18．Reduction Properties

19．Proofs ofthe Reduction Properties in Section 18

20．Evaluation of Reductions

21．The Energy of a Supermartingale Potential

22．The Subtraction of a Supermartingale Discontinuity

23．Supermartingale Decompositions and Discontinuities

Chapter Ⅴ Lattices and Related Classes of Stochastic Processes

1．Conventions;The Essential Order

2．LMx(·)when{x(·),F(·)}Is a Submartingale

3．Uniformly Integrable Positive Submartingales

4．LpBounded Stochastic Processes(p≥1)

5．The Lattices('S±,≤),('S+,≤),(S±,≤),(S+,≤)

6．The Vector Lattices('S,≤)and(S,≤)

7．The Vector Lattices('Sm,≤)and(Sm,≤)

8．The Vector Lattices('Sp,≤)and(Sp,≤)

9．The Vector Lattices('Sqb,≤)and(Sqb,≤)

10．The Vector Lattices('Ss,≤)and(Ss,≤)

11．The Orthogonal Decompositions'Sm='Smqb+'Sms and Sm=Smqb+Sms

12．Local Martingales and Singular Supermartingale Potentials in(S,≤)

13．Quasimartingales(Continuous Parameter Context)

Chapter Ⅵ Markov Processes

1．The Markov Property

2．Choice of Filtration

3．Integral Parameter Markov Processes with Stationary Transition Probabilities

4．Application of Martingale Theory to Discrete Parameter Markov Processes

5．Continuous Parameter Markov Processes with Stationary Transition Probabilities

6．Specialization to Right Continuous Processcs

7．Continuous Parameter Markov Processes:Lifctimes and Trap Points

8．Right Continuity of Markov Process Filtrations;A Zero-One(0-1)Law

9．Strong Markov Property

10．Probabilistic Potential Theory;Excessive Functions

11．Excessive Functions and Supermartingales

12．Excessive Functions and the Hitting Times of Analytic Sets(Notation and Hypotheses of Section 11)

13．Conditioned Markov Processes

14．Tied Down Markov Processes

15．Killed Markov Processes

Chapter Ⅵ Brownian Motion

1．Processes with Independent Increments and State Space RN

2．Brownian Motion

3．Continuity of Brownian Paths

4．Brownian Motion Filtrations

5．Elementary Properties of the Brownian Transition Density and Brownian Motion

6．The Zero-One Law for Brownian Motion

7．Tied Down Brownian Motion

8．André Reflection Principle

9．Brownian Motion in an Open Set(N≥1)

10．Space-Time Brownian Motion in an Open Set

11．Brownian Motion in an Intervai

12．Probabilistic Evaluation of Parabolic Measure for an Interval

13．Probabilistic Significance of the Heat Equation and Its Dual

Chapter Ⅷ

The It？ Integral

1．Notation

2．The Size of Г0

3．Properties ofthe It？ Integral

4．The Stochastic Integral for an Integrand Process in Г0

5．The Stochastic Integral for an Integrand Process in Г

6．Proofs of the Properties in Section 3

7．Extension to Vector-Valued and Complex-Valued Integrands

8．Martingales Relative to Brownian Motion Filtrations

9．A Change of Variables

10．The Role of Brownian Motion Increments

11．(N=1)Computation of the It？ Integral by Riemann-Stieltjes Sums

12．It？'s Lemma

13．The Composition of the Basic Functions of Potential Theory with Brownian Motion

14．The Composition of an Analytic Function with Brownian Motion

Chapter Ⅸ Brownian Motion and Martingale Theory

1．Elementary Martingale Applications

2．Coparabolic Polynomials and Martingale Theory

3．Superharmonic and Harmonic Functions on RN and Supermartingales and Martingales

4．Hitting of an Fσ Set

5．The Hitting of a Set by Brownian Motion

6．Superharmonic Functions,Excessive for Brownian Motion

7．Preliminary Treatment of the Composition of a Superharmonic Function with Brownian Motion;A Probabilistic Fatou Boundary Limit Theorem

8．Excessive and Invariant Functions for Brownian Motion

9．Application to Hitting Probabilities and to Parabolicity of Transition Densities

10．(N=2).The Hitting of Nonpolar Sets by Brownian Motion

11．Continuity of the Composition of a Function with Brownian Motion

12．Continuity of Superharmonic Functions on Brownian Motion

13．Preliminary Probabilistic Solution of the Classical Dirichlet Problem

14．Probabilistic Evaluation of Reductions

15．Probabilistic Description of the Fine Topology

16．α-Excessive Functions for Brownian Motion and Their Composition with Brownian Motions

17．Brownian Motion Transition Functions as Green Functions;The Corresponding Backward and Forward Parabolic Equations

18．Excessive Measures for Brownian Motion

19．Nearly Borel Sets for Brownian Motion

20．Brownian Motion into a Set from an Irregular Boundary Point

Chapter Ⅹ Conditional Brownian Motion

1．Definition

2．h-Brownian Motion in Terms of Brownian Motion

3．Contexts for(2.1)

4．Asymptotic Character of h-Brownian Paths at Their Lifetimes

5．h-Brownian Motion from an Infinity of h

6．Brownian Motion under Time Reversal

7．Preliminary Probabilistic Solution of the Dirichlet Problem for h-Harmonic Functions;h-Brownian Motion Hitting Probabilities and the Corresponding Generalized Reductions

8．Probabilistic Boundary Limit and Internal Limit Theorems for Ratios of Strictly Positive Superharmonic Functions

9．Conditional Brownian Motion in a Ball

10．Conditional Brownian Motion Last Hitting Distributions;The Capacitary Distribution of a Set in Terms of a Last Hitting Distribution

11．The TailσAlgebra ofa Conditional Brownian Motion

12．Conditional Space-Time Brownian Motion

13．[Space-Time]Brownian Motion in[？N]RN with Parameter Set R

Part3

Chapter Ⅰ Lattices in Classical Potential Theory and Martingale Theory

1．Correspondence between Classical Potential Theory and Martingale Theory

2．Relations between Decomposition Components of S in Potential Theory and Martingale Theory

3．The Classes Lp and D

4．PWB-Related Conditions on h-Harmonic Functions and on Martingales

5．Class D Property versus Quasi-Boundedness

6．A Condition for Quasi-Boundedness

7．Singularity of an Element of S+ m

8．The Singular Component of an Element of S+

9．The Class Spqb

10．The Class Sps

11．Lattice Theoretic Analysis of the Composition of an h-Superharmonic Function with an h-Brownian Motion

12．A Decomposition of S+ ms(Potential Theory Context)

13．Continuation of Section 11

Chapter Ⅱ Brownian Motion and the PWB Method

1．Context of the Problem

2．Probabilistic Analysis of the PWB Method

3．PWBh Examples

4．Tail σ Algebras in the PWBh Context

Chapter Ⅲ Brownian Motion on the Martin Space

1．The Structure of Brownian Motion on the Martin Space

2．Brownian Motions from Martin Boundary Points(Notation of Section 1)

3．The Zero-One Law at a Minimal Martin Boundary Point and the Probabilistic Formulation of the Minimal-Fine Topology(Notation of Section 1)

4．The Probabilistic Fatou Theorem on the Martin Space

5．Probabilistic Approach to Theorem 1.XI.4(c)and Its Boundary Counterparts

6．Martin Representation of Harmonic Functions in the Parabolic Context

Appendixes

Appendix Ⅰ Analytic Sets

1．Pavings and Algebras of Sets

2．Suslin Schemes

3．Sets Analytic over a Product Paving

4．Analytic Extensions versus σ Algebra Extensions of Pavings

5．Projection Characterization A(y)

6．The Operation A(A)

7．Projections of Sets in Product Pavings

8．Extension of a Measurability Concept to the Analytic Operation Context

9．The Gδ Sets of a Complete Metric Space

10．Polish Spaces

11．The Baire Null Space

12．Analytic Sets

13．Analytic Subsets of Polish Spaces

Appendix Ⅱ Capacity Theory

1．Choquet Capacities

2．Sierpinski Lemma

3．Choquet Capacity Theorem

4．Lusin's Theorem

5．A Fundamental Example of a Choquet Capacity

6．Strongly Subadditive Set Functions

7．Generation of a Choquet Capacity by a Positive Strongly Subadditive Set Function

8．Topological Precapacities

9．Universally Measurable Sets

Appendix Ⅲ Lattice Theory

1．Introduction

2．Lattice Definitions

3．Cones

4．The Specific Order Generated by a Cone

5．Vector Lattices

6．Decomposition Property of a Vector Lattice

7．Orthogonality in a Vector Lattice

8．Bands in a Vector Lattice

9．Projections on Bands

10．The Orthogonal Complement of a Set

11．The Band Generated by a Single Element

12．Order Convergence

13．Order Convergence on a Linearly Ordered Set

Appendix Ⅳ Lattice Theoretic Concepts in Measure Theory

1．Lattices of Set Algebras

2．Measurable Spaces and Measurable Functions

3．Composition of Functions

4．The Measure Lattice of a Measurable Space

5．The σ Finite Measure Lattice of a Measurable Space(Notation of Section 4)

6．The Hahn and Jordan Decompositions

7．The Vector Lattice Mσ

8．Absolute Continuity and Singularity

9．Lattices of Measurable Functions on a Measure Space

10．Order Convergence of Families of Measurable Functions

11．Measures on Polish Spaces

12．Derivates of Measures

Appendix Ⅴ Uniform Integrability

Appendix Ⅵ Kernels and Transition Functions

1．Kernels

2．Universally Measurable Extension of a Kernel

3．Transition Functions

Appendix Ⅶ Integral Limit Theorerns

1．An Elementary Limit Theorem

2．Ratio Integral Limit Theorems

3．A One-Dimensional Ratio Integral Limit Theorem

4．A Ratio Integral Limit Theorem Involving Convex Variational Derivates

Appendix Ⅷ Lower Semicontinuous Functions

1．The Lower Sernicontinuous Smoothing of a Function

2．Suprema of Families of Lower Semicontinuous Functions

3．Choquet Topological Lemma

Historical Notes

Part 1

Part 2

Part 3

Appendixes

Bibliography

Notation Index

Index