安裝中文字典英文字典辭典工具!
安裝中文字典英文字典辭典工具!
|
- Measure Spaces - UC Davis
We say that a measure space (X,A,µ) is finite if the set X has finite measure, and σ-finite if X is a countable union of sets of finite measure For example, any probablity space, which is a measure space such that µ(X) = 1 is finite, and Rd equipped with Lebesgue measure is σ-finite but not finite
- 43. Complete Metric Spaces Chapter 7. Complete Metric Spaces and . . .
Definition Let (X,d) be a metric space A sequence (xn) of points of X is a Cauchy sequence on (X,d) if for all ε > 0 there is N ∈ N such that if m,n ≥ N then d(xn,xm) < ε The metric space (X,d) is complete if every Cauchy sequence in X converges Note By the Triangle Inequality for any metric, a convergent sequence is always
- 10 CHAPTER 1. MEASURE - UH
So ¯µ is a measure To see that it is complete, choose C ∈ A¯ which is ¯µ-null, and D ⊂ C Write C = A ∪ F with A ∈ A,F ⊂ B ∈ A, with µ(B) = 0 Since C is ¯µ-null, µ(A) = 0 Thus D is a subset of the µ-null set A∪ B, so D ∈ A¯ We call ¯µ the completion of µ, and we call A¯ the completion of A with respect to µ
- 2. 3 Basic Properties of Measures - gatech. edu
Theorem 2 24 (Countable Subadditivity) Let (X,Σ,µ) be a measure space If E1,E2, ∈ Σ, then µ S k Ek ≤ X k µ(Ek) Proof Using the disjointization trick (Exercise 2 7), we can write ∪Ek = ∪Fk where the sets Fk are measurable and disjoint, and Fk ⊆ Ek for each k Combining countable additivity and monotonicity, it follows that
- Completion of a measure space - Mathematics Stack Exchange
Studying Measure Theory in University, I came across the following definition for the completion of a measure space: let be a measure space; then the set ¯ E = {A ⊆ X: ∃B, C ∈ E: A B ⊆ C ∧ μ(C) = 0} is a σ -algebra and, extending μ to ¯ μ defined on ¯ E by letting ¯ μ(A) = μ(B) for any set B ∈ E such that A B is contained in a null set, the meas
- Let X,d be a metric space. - University of California, Berkeley
Let (X,d) be a metric space 1) A sequence (x n) in X converges to x if (∀ǫ > 0)(∃N ∈ N) so that (n ≥ N) =⇒ d(x n,x) < ǫ 2) A sequence (x n) is Cauchy if (∀ǫ > 0)(∃N ∈ N) so that (n,m ≥ N) =⇒ d(x n,x m) < ǫ 3) A metric space is said to be complete if every Cauchy sequence converges
- Lecture 4: Completion of a Metric Space - George Mason University
X˜ is a complete metric space Proof of Claim 5: Let fx˜kg1 k=1 be Cauchy in X˜ with ˜x k = [fxk ng] for each k fx˜kg Cauchy means that given † > 0, there exists N such that if k; j ‚ N, then d˜(˜x k;x˜j) = lim n!1 d(xk n;x j) < †: We must show that f˜xkg in fact converges The first step is to define a candidate, ˜x, for the
- Complete Metric Spaces
Definition 1 Let (X, d) be a metric space A sequence (xn) in X is called a Cauchy sequence if for any ε > 0, there is an nε ∈ N such that d(xm, xn) < ε for any m ≥ nε, n ≥ nε Theorem 2 Any convergent sequence in a metric space is a Cauchy sequence Proof Assume that (xn) is a sequence which converges to x Let ε > 0 be ε given
|
|
|