Robert R. Reitano is Professor of the Practice in Finance at the Brandeis International Business School where he specializes in risk management and quantitative finance, and where he previously served as MSF Program Director, and Senior Academic Director. He has a Ph.D. in Mathematics from MIT, is a Fellow of the Society of Actuaries, and a Chartered Enterprise Risk Analyst. He has taught as Visiting Professor at Wuhan University of Technology School of Economics, Reykjavik University School of Business, and as Adjunct Professor in Boston University's Masters Degree program in Mathematical Finance. Dr. Reitano consults in investment strategy and asset/liability risk management, was Chief Investment Officer of Controlled Risk Insurance Company (CRICO), and previously had a 29-year career at John Hancock/Manulife in investment strategy and asset/liability management, advancing to Executive Vice President & Chief Investment Strategist. His research papers have appeared in a number of journals and have won an Annual Prize of the Society of Actuaries and two F.M. Redington Prizes awarded biennially by the Investment Section of the Society of the Actuaries. Dr. Reitano has served as Vice-Chair of the Board of Directors of the Professional Risk Managers International Association (PRMIA) and on the Executive Committee of the PRMIA Board, and is currently a member of the PRMIA Boston Steering Committee, the Financial Research Committee of the Society of Actuaries, and other not-for-profit boards and investment committees.

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• Preface Introduction 1 The Notion of Measure 0 1.1 The Riemann Integral 1.2 The Lebesgue Integral 2 Lebesgue Measure on R 132.1 Sigma Algebras and Borel Sets 2.2 Definition of a Lebesgue Measure 2.3 There is No Lebesgue Measure on _(P(R)2.4 Lebesgue Measurable Sets: ML(R) $ _(P(R)) 2.5 Calculating Lebesgue Measures 2.6 Approximating Lebesgue Measurable Sets 2.7 Properties of Lebesgue Measure 2.7.1 Regularity 2.7.2 Continuity 2.8 Discussion on B(R) &ML(R) 3 Measurable Functions 553.1 Extended Real-Valued Functions 3.2 Equivalent Definitions of Measurability 3.3 Examples of Measurable Functions 3.4 Properties of Measurable Functions 3.4.1 Elementary Function Combinations 3.4.2 Function Sequences Function Sequence Behaviors Function Sequence Measurability Properties 3.5 Approximating Lebesgue Measurable Functions 3.6 Distribution Functions of Measurable Functions 4 Littlewood.s Three Principles 4.1 Measurable Sets 4.2 Convergent Sequences of Measurable Functions 4.3 Measurable Functions 5 Borel Measures on R 5.1 Functions Induced by Borel Measures 5.2 Borel Measures from Distribution Functions 5.3 Consistency of Borel Measure Constructions 5.4 Approximating Borel Measurable Sets5.5 Properties of Borel Measures 5.6 Differentiable F-Length and Lebesgue Measure 6 Generating Measures by Extension 6.1 Recap of Lebesgue and Borel Constructions 6.2 Extension Theorems 6.3 Summary - Construction of Measure Spaces 6.4 Approaches to Countable Additivity 6.5 Completion of a Measure Space7 Finite Products of Measure Spaces 7.1 Product Space Semi-Algebras 7.2 Properties of the Semi-Algebra 7.3 Measure on the Algebra A 7.4 Extension to a Measure on the Product Space 7.5 Well-Definedness of _-Finite Product Measure Spaces 7.6 Products of Lebesgue and Borel Measure Spaces 8 Borel Measures on Rn 8.1 Rectangle Collections that Generate B(Rn) 8.2 Borel Measures and Induced Functions 8.3 Properties of General Borel Measures on Rn 9 Infinite Products of Probability Spaces 9.1 A Naive Attempt at a First Step 9.2 A Semi-Algebra A0 9.3 Finite Additivity of _A on A for Probability Spaces 9.4 Free Countable Additivity on Finite Probability Spaces 9.5 Countable Additivity on A+ in Probability Spaces on R 9.6 Extension to a Probability Measure on RN 9.7 Probability of General Rectangles References

Robert R. Reitano is Professor of the Practice in Finance at the Brandeis International Business School where he specializes in risk management and quantitative finance, and where he previously served as MSF Program Director, and Senior Academic Director. He has a Ph.D. in Mathematics from MIT, is a Fellow of the Society of Actuaries, and a Chartered Enterprise Risk Analyst. He has taught as Visiting Professor at Wuhan University of Technology School of Economics, Reykjavik University School of Business, and as Adjunct Professor in Boston University's Masters Degree program in Mathematical Finance. Dr. Reitano consults in investment strategy and asset/liability risk management, was Chief Investment Officer of Controlled Risk Insurance Company (CRICO), and previously had a 29-year career at John Hancock/Manulife in investment strategy and asset/liability management, advancing to Executive Vice President & Chief Investment Strategist. His research papers have appeared in a number of journals and have won an Annual Prize of the Society of Actuaries and two F.M. Redington Prizes awarded biennially by the Investment Section of the Society of the Actuaries. Dr. Reitano has served as Vice-Chair of the Board of Directors of the Professional Risk Managers International Association (PRMIA) and on the Executive Committee of the PRMIA Board, and is currently a member of the PRMIA Boston Steering Committee, the Financial Research Committee of the Society of Actuaries, and other not-for-profit boards and investment committees.

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• Preface Introduction 1 The Notion of Measure 0 1.1 The Riemann Integral 1.2 The Lebesgue Integral 2 Lebesgue Measure on R 132.1 Sigma Algebras and Borel Sets 2.2 Definition of a Lebesgue Measure 2.3 There is No Lebesgue Measure on _(P(R)2.4 Lebesgue Measurable Sets: ML(R) $ _(P(R)) 2.5 Calculating Lebesgue Measures 2.6 Approximating Lebesgue Measurable Sets 2.7 Properties of Lebesgue Measure 2.7.1 Regularity 2.7.2 Continuity 2.8 Discussion on B(R) &ML(R) 3 Measurable Functions 553.1 Extended Real-Valued Functions 3.2 Equivalent Definitions of Measurability 3.3 Examples of Measurable Functions 3.4 Properties of Measurable Functions 3.4.1 Elementary Function Combinations 3.4.2 Function Sequences Function Sequence Behaviors Function Sequence Measurability Properties 3.5 Approximating Lebesgue Measurable Functions 3.6 Distribution Functions of Measurable Functions 4 Littlewood.s Three Principles 4.1 Measurable Sets 4.2 Convergent Sequences of Measurable Functions 4.3 Measurable Functions 5 Borel Measures on R 5.1 Functions Induced by Borel Measures 5.2 Borel Measures from Distribution Functions 5.3 Consistency of Borel Measure Constructions 5.4 Approximating Borel Measurable Sets5.5 Properties of Borel Measures 5.6 Differentiable F-Length and Lebesgue Measure 6 Generating Measures by Extension 6.1 Recap of Lebesgue and Borel Constructions 6.2 Extension Theorems 6.3 Summary - Construction of Measure Spaces 6.4 Approaches to Countable Additivity 6.5 Completion of a Measure Space7 Finite Products of Measure Spaces 7.1 Product Space Semi-Algebras 7.2 Properties of the Semi-Algebra 7.3 Measure on the Algebra A 7.4 Extension to a Measure on the Product Space 7.5 Well-Definedness of _-Finite Product Measure Spaces 7.6 Products of Lebesgue and Borel Measure Spaces 8 Borel Measures on Rn 8.1 Rectangle Collections that Generate B(Rn) 8.2 Borel Measures and Induced Functions 8.3 Properties of General Borel Measures on Rn 9 Infinite Products of Probability Spaces 9.1 A Naive Attempt at a First Step 9.2 A Semi-Algebra A0 9.3 Finite Additivity of _A on A for Probability Spaces 9.4 Free Countable Additivity on Finite Probability Spaces 9.5 Countable Additivity on A+ in Probability Spaces on R 9.6 Extension to a Probability Measure on RN 9.7 Probability of General Rectangles References

Robert R. Reitano is Professor of the Practice in Finance at the Brandeis International Business School where he specializes in risk management and quantitative finance, and where he previously served as MSF Program Director, and Senior Academic Director. He has a Ph.D. in Mathematics from MIT, is a Fellow of the Society of Actuaries, and a Chartered Enterprise Risk Analyst. He has taught as Visiting Professor at Wuhan University of Technology School of Economics, Reykjavik University School of Business, and as Adjunct Professor in Boston University's Masters Degree program in Mathematical Finance. Dr. Reitano consults in investment strategy and asset/liability risk management, was Chief Investment Officer of Controlled Risk Insurance Company (CRICO), and previously had a 29-year career at John Hancock/Manulife in investment strategy and asset/liability management, advancing to Executive Vice President & Chief Investment Strategist. His research papers have appeared in a number of journals and have won an Annual Prize of the Society of Actuaries and two F.M. Redington Prizes awarded biennially by the Investment Section of the Society of the Actuaries. Dr. Reitano has served as Vice-Chair of the Board of Directors of the Professional Risk Managers International Association (PRMIA) and on the Executive Committee of the PRMIA Board, and is currently a member of the PRMIA Boston Steering Committee, the Financial Research Committee of the Society of Actuaries, and other not-for-profit boards and investment committees.

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• Preface Introduction 1 The Notion of Measure 0 1.1 The Riemann Integral 1.2 The Lebesgue Integral 2 Lebesgue Measure on R 132.1 Sigma Algebras and Borel Sets 2.2 Definition of a Lebesgue Measure 2.3 There is No Lebesgue Measure on _(P(R)2.4 Lebesgue Measurable Sets: ML(R) $ _(P(R)) 2.5 Calculating Lebesgue Measures 2.6 Approximating Lebesgue Measurable Sets 2.7 Properties of Lebesgue Measure 2.7.1 Regularity 2.7.2 Continuity 2.8 Discussion on B(R) &ML(R) 3 Measurable Functions 553.1 Extended Real-Valued Functions 3.2 Equivalent Definitions of Measurability 3.3 Examples of Measurable Functions 3.4 Properties of Measurable Functions 3.4.1 Elementary Function Combinations 3.4.2 Function Sequences Function Sequence Behaviors Function Sequence Measurability Properties 3.5 Approximating Lebesgue Measurable Functions 3.6 Distribution Functions of Measurable Functions 4 Littlewood.s Three Principles 4.1 Measurable Sets 4.2 Convergent Sequences of Measurable Functions 4.3 Measurable Functions 5 Borel Measures on R 5.1 Functions Induced by Borel Measures 5.2 Borel Measures from Distribution Functions 5.3 Consistency of Borel Measure Constructions 5.4 Approximating Borel Measurable Sets5.5 Properties of Borel Measures 5.6 Differentiable F-Length and Lebesgue Measure 6 Generating Measures by Extension 6.1 Recap of Lebesgue and Borel Constructions 6.2 Extension Theorems 6.3 Summary - Construction of Measure Spaces 6.4 Approaches to Countable Additivity 6.5 Completion of a Measure Space7 Finite Products of Measure Spaces 7.1 Product Space Semi-Algebras 7.2 Properties of the Semi-Algebra 7.3 Measure on the Algebra A 7.4 Extension to a Measure on the Product Space 7.5 Well-Definedness of _-Finite Product Measure Spaces 7.6 Products of Lebesgue and Borel Measure Spaces 8 Borel Measures on Rn 8.1 Rectangle Collections that Generate B(Rn) 8.2 Borel Measures and Induced Functions 8.3 Properties of General Borel Measures on Rn 9 Infinite Products of Probability Spaces 9.1 A Naive Attempt at a First Step 9.2 A Semi-Algebra A0 9.3 Finite Additivity of _A on A for Probability Spaces 9.4 Free Countable Additivity on Finite Probability Spaces 9.5 Countable Additivity on A+ in Probability Spaces on R 9.6 Extension to a Probability Measure on RN 9.7 Probability of General Rectangles References