Let ((Omega, mathscr{A}, mathbb{P})) be a probability space. Adapt the proof of Theorem 22.9 to show that
Question:
Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space. Adapt the proof of Theorem 22.9 to show that a sequence \(\left(u_{n}ight)_{n \in \mathbb{N}} \subset \mathcal{L}^{1}(\mu)\) is uniformly integrable if it is bounded in some space \(\mathcal{L}^{p}(\mathbb{P})\) with \(p>1\), i.e. if \(\sup _{n \in \mathbb{N}}\left\|u_{n}ight\|_{p}
Use Vitali's convergence theorem to construct an example illustrating that \(\mathcal{L}^{1}\) boundedness of \(\left(u_{n}ight)_{n \in \mathbb{N}}\) does not guarantee uniform integrability.
Data from theorem 22.9
Transcribed Image Text:
Theorem 22.9 Let (X, A, ) be some measure space and FCL(A). Then the following statements (i)-(v) are equivalent: (i) F is uniformly integrable, i.e. (22.1) holds; (ii) (a) supu d 0 WEL (A), 5>0 VBA, (iii) (a) supu Swee We d < 6: sup / \u\ d < ; B uEJ JB d 0 KEA, H(K) 0 VBA, (B) <8: sup 2 | Judul <6; uEJ B (iv) (a) supu d 0KA, (K) 0_\8>0\B(B) <8 : sup | \u\d 0 KEA, (K) R} uEJ J K
Fantastic news! We've Found the answer you've been seeking!
Step by Step Answer:
Answer rating: 100% (4 reviews)
P we get Since the function u R R 0 is integrable wrt the probability measure 1412 14 1 dp ujR P ...View the full answer
Answered By
Ashington Waweru
I am a lecturer, research writer and also a qualified financial analyst and accountant. I am qualified and articulate in many disciplines including English, Accounting, Finance, Quantitative spreadsheet analysis, Economics, and Statistics. I am an expert with sixteen years of experience in online industry-related work. I have a master's in business administration and a bachelor’s degree in education, accounting, and economics options.
I am a writer and proofreading expert with sixteen years of experience in online writing, proofreading, and text editing. I have vast knowledge and experience in writing techniques and styles such as APA, ASA, MLA, Chicago, Turabian, IEEE, and many others.
I am also an online blogger and research writer with sixteen years of writing and proofreading articles and reports. I have written many scripts and articles for blogs, and I also specialize in search engine
I have sixteen years of experience in Excel data entry, Excel data analysis, R-studio quantitative analysis, SPSS quantitative analysis, research writing, and proofreading articles and reports. I will deliver the highest quality online and offline Excel, R, SPSS, and other spreadsheet solutions within your operational deadlines. I have also compiled many original Excel quantitative and text spreadsheets which solve client’s problems in my research writing career.
I have extensive enterprise resource planning accounting, financial modeling, financial reporting, and company analysis: customer relationship management, enterprise resource planning, financial accounting projects, and corporate finance.
I am articulate in psychology, engineering, nursing, counseling, project management, accounting, finance, quantitative spreadsheet analysis, statistical and economic analysis, among many other industry fields and academic disciplines. I work to solve problems and provide accurate and credible solutions and research reports in all industries in the global economy.
I have taught and conducted masters and Ph.D. thesis research for specialists in Quantitative finance, Financial Accounting, Actuarial science, Macroeconomics, Microeconomics, Risk Management, Managerial Economics, Engineering Economics, Financial economics, Taxation and many other disciplines including water engineering, psychology, e-commerce, mechanical engineering, leadership and many others.
I have developed many courses on online websites like Teachable and Thinkific. I also developed an accounting reporting automation software project for Utafiti sacco located at ILRI Uthiru Kenya when I was working there in year 2001.
I am a mature, self-motivated worker who delivers high-quality, on-time reports which solve client’s problems accurately.
I have written many academic and professional industry research papers and tutored many clients from college to university undergraduate, master's and Ph.D. students, and corporate professionals. I anticipate your hiring me.
I know I will deliver the highest quality work you will find anywhere to award me your project work. Please note that I am looking for a long-term work relationship with you. I look forward to you delivering the best service to you.
3.00+
2+ Reviews
10+ Question Solved
Related Book For
Question Posted:
Students also viewed these Business questions
-
If one attempted to dualize the notion of free module over a ring R (and called the object so defined "co-free") the definition would read: An R-module F is co-free on a set X if there exists a...
-
Let \((X, d)\) be a locally compact metric space and \(\mu, \mu_{n} \in \mathfrak{M}_{\mathrm{r}}^{+}(X), \mu_{n} \stackrel{\mathrm{v}}{ightarrow} \mu\). Prove that \[\lim _{n} \int_{B} u d...
-
Completing a detail of the proof of Theorem 12.5, let G be a finite group of plane isometries. Show that the rotations in G, together with the identity isometry, form a subgroup H of G, and that...
-
The HaberBosch process for the production of ammonia is one of the key industrial processes in developed countries. N 2 (g) + 3 H 2 (g) 2 NH 3 (g) (a) Calculate r G for the reaction at 298 K, 800...
-
An electron in the n = 2 state of hydrogen remains there on average about 10 8s before jumping to the n = 1 state. (a) Estimate the uncertainty in the energy of the n = 3 state. (b) What fraction of...
-
Each of the following compounds does not participate as a diene in a Diels-Alder reaction. Explain why in each case. (a) (b) (c) (d)
-
Whether hearsay evidence may be admissible under a hearsay exception?
-
Calculating the materials usage variance Laura Pierre is the manager of the Roebuck Bagel Shop. The corporate office had budgeted her store to sell 3,000 ham sandwiches during the week beginning July...
-
Customer service: The Role of Gender in Nonverbal Communication Using this link: https://freemanmeansbusiness.com/blog/2018/9/13/gender-differences-in-communication-styles Find: Brain Differences...
-
Let \((X, \mathscr{A}, \mu)\) be a finite measure space. Show that \(\mathcal{F} \subset \mathcal{L}^{1}(\mu)\) is uniformly integrable if, and only if, the series \(\sum_{n=1}^{\infty} n \mu\{n [...
-
Assume that \(\left(u_{n}ight)_{n \in \mathbb{N}}\) is uniformly integrable. Show that \[\lim _{k ightarrow \infty} \frac{1}{k} \int \max _{n \leqslant k} u_{n} d \mu=0\]
-
To what potential should you charge a 1.0 F capacitor to store 1.0 J of energy?
-
In 2 0 2 4 , due to a change in marketing forecasts, Barney Corporation reduced the projected life of its patent for producing round dice. The cumulative patent amortization prior to 2 0 2 4 would...
-
Cost effectiveness assignment For this assignment, you are being asked to evaluate the cost-benefit of one of four public health issues. You will be assigned to one of the following (see Week 3...
-
You are inside a spaceship in deep space, far away from any stars. Your engines are off and you believe that your spaceship is stationary. You look out and see Phil and Victor approaching you from...
-
Solve the following question from the data given below. 4 1.9 7.8 16 0.71 2 [1.22 X X 43 #1719 Similar question X 04 08 y 800 980 8.7 solve using Gauss- Jordan solve using LU Decomposition 2 2.3...
-
The general management knowledge and practice competency area includes knowledge and skills such as: 1. risk management, schedule development, cost estimating and budgeting, and requirements...
-
Winters Co. purchased $48,000 of marketable securities on March 1, 2013. On the companys fiscal year closing date, December 31, 2013, the securities had a market value of $27,000. During 2013,...
-
A copper wire (density = 8.96 g/cm 3 ) has a diameter of 0.25 mm. If a sample of this copper wire has a mass of 22 g, how long is the wire?
-
Why are the terms product costs and period costs used to describe manufacturing costs and non-manufacturing costs?
-
How does the timing of recording expenses differ between product and period costs?
-
Why are items such as the hull, engine, transmission, carpet, and seats classified as direct materials and items such as glue, paint, and screws classified as indirect materials?
-
The financial statements of Inland Engineering are provided below. The firm has 100 million shares outstanding at the end of 2020 and the stock price was $10. The firm plans to increase its fixed...
-
Find the entire domain on which the function f is one-to-one and non-decreasing. Write the domain in interval notation. f(x) = (x + 1) Find the inverse of f restricted to that domain. F-1(x) = Submit...
-
Consider the following initial-value problem. f'(x) = 2ex - 6x; f(0) = 4 Integrate the function f'(x). (Remember the constant of integration.) || | f'(x)dx = Find the value of C using the condition...
Study smarter with the SolutionInn App