Slides Chapter 1. Measure Theory and Probability

More documents

Recommendations

Info

1.3. SET FUNCTIONS Example 1.5 (Counting measure) Let Ω be any set and consider the σ-algebra of the parts of Ω, P(Ω). Define µ(A) as the number of points of A. The set function µ is a measure known as the counting measure. Example 1.6 (Probability measure) Let Ω = {x 1 ,x 2 ,...} be a finite or countably infinite set, and let p 1 ,p 2 ,..., be nonnegative numbers. Consider the σ-algebra of the parts of Ω, P(Ω), and define µ(A) = ∑ x i ∈Ap i . Thesetfunctionµisaprobabilitymeasureifandonlyif ∑ ∞ i=1 p i = 1. Example 1.7 (Lebesgue measure) A well known measure defined over (IR,B), which assigns to each element of B its length, is the Lebesgue measure, denoted here as λ. For an interval, either open, close or semiclosed, the Lebesgue measure is the length of the interval. For a single point, the Lebesgue measure is zero. Definition 1.16 (σ-finite set function) A set function φ : A → IR is σ-finite if ∀A ∈ A, there exists a sequence {A n } n∈IN of disjoint elements of A with φ(A n ) < ∞ ∀n, whose union covers A, that is, ∞⋃ A ⊆ A n . n=1 Definition 1.17 (Measure space) The triplet (Ω,A,µ), where Ω is a sample space, A is an algebra and µ is a measure defined over (Ω,A), is called measure space. ISABEL MOLINA 10
1.3. SET FUNCTIONS Definition 1.18 (Absolutely continuous measure with respect to another) A measure µ on Borel subsets of the real line is absolutely continuous with respect to another measure λ if λ(A) = 0 implies that µ(A) = 0. It is also said that µ is dominated by λ, and written as µ
Page 1 and 2: Chapter 1 Measure theory and Probab
Page 3 and 4: 1.1. SET SEQUENCES (ii) limA n =
Page 5 and 6: 1.2. STRUCTURES OF SUBSETS 8. Let{x
Page 7 and 8: 1.2. STRUCTURES OF SUBSETS σ-algeb
Page 9: 1.3. SET FUNCTIONS Definition 1.12
Page 13 and 14: 1.4. PROBABILITY MEASURES (b) Obser
Page 15 and 16: 1.4. PROBABILITY MEASURES (i) P(∅
Page 17 and 18: 1.4. PROBABILITY MEASURES For this,
Page 19 and 20: 1.5. OTHER DEFINITIONS OF PROBABILI
Page 25 and 26: 1.6. MEASURABILITY AND LEBESGUE INT
Page 31 and 32: 1.7. DISTRIBUTION FUNCTION Proposit
Page 33 and 34: 1.7. DISTRIBUTION FUNCTION Definiti
Page 35 and 36: 1.8. RANDOM VARIABLES (b) If f is c
Page 37 and 38: 1.8. RANDOM VARIABLES (b) For the m
Page 39 and 40: 1.8. RANDOM VARIABLES Summarizing,
Page 41 and 42: 1.8. RANDOM VARIABLES We will also
Page 43 and 44: 1.8. RANDOM VARIABLES (b) F X absol
Page 45 and 46: 1.9. THE CHARACTERISTIC FUNCTION Th
Page 47 and 48: 1.9. THE CHARACTERISTIC FUNCTION Pr
Page 49 and 50: 1.9. THE CHARACTERISTIC FUNCTION Th
Page 51 and 52: 1.9. THE CHARACTERISTIC FUNCTION Co
Page 53 and 54: 1.9. THE CHARACTERISTIC FUNCTION (b

Slides Chapter 1. Measure Theory and Probability

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?