Preliminaries on Measure and Integration in Lp-space

1. Chapter 3 Lp-space

2. Preliminaries on measure and integration

3. σ-algebra Ω ≠ψ is a set Σis a family of subsets of Ω with Σis called σ-algebra of subsets of Ω

4. measure space Ω ≠ψ is a set Σis aσ-algebra of subsets of Ω μ:Σ→[0, ∞] satisfies (Ω , Σ, μ) is called a measure space

5. measurable function p.1 (Ω , Σ, μ) is a measure space f:Ω→R is measurable if The family of measurable functions is a real vector space.

6. measurable function p.2 The family isclosed under limit, i.e. if is a sequence of measurable functions , which converges pointwise to a finite-valued function f , then f is measurable.(see Exercise 1.1 and 1.3)

7. Exercise 1.1 (Ω , Σ, μ) is a measure space If f, g are measurable , then f+g is also measurable. Hint: for all

10. Exercise 1.3 (Ω , Σ, μ) is a measure space Let f1,f2,… be measurable and and f(x) is finite for each Show that f is measurable Hint: for all

13. indicator function (Ω , Σ, μ) is a measure space for χA is the indicator function of A χA is measurable

14. <W> <W> denotes the smallest vector subspace containing W in a vector space.

15. Simple function p.1 Elements of are called simple functions

16. Simple function p.2 A simple function can be expressed as , where are different values andAi = {f =αi} , we define then if the right hand side has a meaning

17. Simple function p.3 In particular is meaningful if f is simple and nonnegative, although it is possiblethat

18. Integration for f≧0,measurable For f≧0, measurable ,define

19. f+ ,f- If f is measurable, then f+ and f- are measurable

20. Integration for measurable function p.1 For any measurable function f ,define if R.H.S has a meaning

21. Integration for measurable function p.2 is finite if and only if both are finite and f is called integrable

22. Integration for measurable function p.3 f is integrable if and only if is integrable

23. limsupAn , liminf An Ω is a set and {An} is a sequence of subsets of Ω. Define

24. limAn If then we say that the limit of the sequence {An} exists and has the common set as the limit which is denoted by

25. An: monotone increasing If then

26. An: monotone decreasing If then

27. Lemma 2.1 (Ω , Σ, μ) is a measure space If be monotone increasing, then

29. Lemma 2.2 (Ω , Σ, μ) is a measure space If be monotone decreasing, then

31. Egoroff Theorem (Ω , Σ, μ) is a measure space Let {fn} be a sequence of measurable function and fn→f with finite limit on then for any ε>0 , there is such that μ(A\B)<ε and fn→f uniformly on B

34. Monotony Convergence Theorem (Ω , Σ, μ) is a measure space Let {fn} be a nondecreasing sequence of nonnegative measurable functions Suppose is a finite valued,then

35. Theorem(Beppo-Levi) (Ω , Σ, μ) is a measure space Let {fn} be a increasing sequence of integrable functions such that fn↗f . Then f is integrable and

37. Fatous Lemma (Ω , Σ, μ) is a measure space Let {fn} be a sequence of extended real-valued measurable functions which is bounded from below by an integrable function. Then

39. Remark (Ω , Σ, μ) is a measure space Let {fn} be a sequence of extended real-valued measurable functions and fn≦0. Then

41. Lebesque Dominated Convergence Theorem (Ω , Σ, μ) is a measure space If fn ,n=1,2,…, and f are measurable functions and fn →f a.e. Suppose that a.e. with g being an integrable function.Then

43. Corollary (Ω , Σ, μ) is a measure space If fn ,n=1,2,…, and f are measurable functions and fn →f a.e. Suppose that a.e. with g being an integrable function.Then

45. 5 The space Lp(Ω,Σ,μ)

46. For measurable function f, let is called the essential sup-norm of f.

47. Exercise 5.1 Show that

49. Conjugate exponents If are such that then they are called conjugate exponents

50. Theorem 5.1(Hölder’s Inequality) p.1 If are conjugate exponents, then