Transcription of Chapter 5. Measurable Functions 1. Measurable Functions
{{id}} {{{paragraph}}}
Example: tourism industry
Chapter 5. Measurable Functions 1. Measurable Functions
Chapter 5. Measurable Functions 1. Measurable FunctionsLetXbe a nonempty set, and letSbe a -algebra of subsets ofX. Then (X, S) is ameasurable space. A subsetEofXis said to be Measurable ifE this Chapter , we will consider Functions fromXtoIR, whereIR := IR { } {+ }is the set ofextended real numbers. For simplicity, we write for + . The setIR isan ordered set: < x < forx functionffromXtoIR is calledmeasurableif, for eacha IR,{x X:f(x)> a}is a Measurable a function from a Measurable space(X, S)toIR. Then thefollowing conditions are equivalent:(1)fis Measurable ;(2) for eacha IR,f 1([a, ])is Measurable ;(3) for eacha IR,f 1([ , a))is Measurable ;(4) for eacha IR,f 1([ , a])is Measurable ;(5) the setsf 1({ })andf 1({ })are Measurable , and for each pair of real numbersaandbwitha < b,f 1((a, b))is (1) (2):f 1([a, ]) = n=1f 1((a 1/n, ]).(2) (3):f 1([ , a)) =X\f 1([a, ]).(3) (4):f 1([ , a]) = n=1f 1([ , a+ 1/n)).(4) (5): If (4) is true, thenf 1([ , b)) = n=1f 1([ , b 1/n]) is measur-able.]]]
Chapter 5. Measurable Functions §1. Measurable Functions Let X be a nonempty set, and let S be a σ-algebra of subsets of X. Then (X,S) is a measurable space. A subset E of X is said to be measurable if E ∈ S. In this chapter, we will consider functions from X to IR, where IR := IR∪{−∞}∪{+∞} is the set of extended real numbers.
Tags:
Information
Domain:
Source:
Link to this page:
Please notify us if you found a problem with this document:
Related search queries
Chapter, Relations and Functions Multiple Choice Questions 1, Chapter 5 Special Functions, Chapter 5 SPECIAL FUNCTIONS Chapter 5 SPECIAL FUNCTIONS, Functions, CHAPTER 10 Limits of Trigonometric Functions, Chapter 1 Character Functions, Chapter 1, Character Functions, Character, SAS character functions, Chapter 4 Airy Functions, JOINT PROBABILITY, Probability