Let ((X, mathscr{A}, mu)) be a finite measure space and let (u in mathcal{L}^{1}(mu)) be strictly positive

Question:

Let $(X, \mathscr{A}, \mu)$ be a finite measure space and let $u \in \mathcal{L}^{1}(\mu)$ be strictly positive with $\int u d \mu=1$. Show that

\[\int(\log u) d \mu \leqslant \mu(X) \log \left(\frac{1}{\mu(X)}ight)\]

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For book-img-for-question

Measures Integrals And Martingales

ISBN: 9781316620243

2nd Edition

Authors: René L. Schilling

See More Books

Question Posted: Jan 25, 2024 01:02 AM

See More Questions

. Let $(X, \mathscr{A}, \mu)$ be a finite measure space and $\left(A_{n}ight)_{n \in \mathbb{N}},\left(B_{n}ight)_{n \in \mathbb{N}} \subset \mathscr{A}$ such that $A_{n} \supset B_{n}$ for all...
Stielties measure (3). Let $(\mathbb{R}, \mathscr{B}(\mathbb{R}), \mu)$ be a finite measure space and denote by $F$ the left-continuous distribution function of $\mu$ as in Problem 6.1. Use...
Let $(X, \mathscr{A}, \mu)$ be a general measure space and let $u \in \bigcap_{p \geqslant 1} \mathcal{L}^{p}(\mu)$. Then \[\lim _{p ightarrow \infty}\|u\|_{p}=\|u\|_{\infty}\] where...
Please complete the Web case that covers material discussed in this chapter at www.cengage.com/accounting/Gibson. Youll be using Thomson ONE Business School Edition, a powerful tool that combines a...
(a) Show that the density of nuclear matter is essentially the same for all nuclei. (b) What would be the radius of the Earth if it had its actual mass but had the density of nuclei? (c) What would...
Are there any cultural practices that are reprehensible in your culture? Are there any Australian cultural practices that may be difficult for a foreigner to understand? How would you deal with this...
Have you been feeling uncomfortably tense lately? True/False LO.1
On March 1, 2015, Zephur Winds Ltd. purchased a machine for $80,000 by paying $20,000 down and issuing a note for the balance. The machine had an estimated useful life of nine years and an estimated...
HBB Company for the last ten years, has earned and had cash flows of about Php 500,000 every year. As per the predictions of the company's earnings, the same cash flow would continue for the...
Let $u$ be a positive measurable function on $[0,1]$. Which of the following is larger: \[\int_{(0,1)} u(x) \log u(x) \lambda(d x) \quad \text { or } \quad \int_{(0,1)} u(s) \lambda(d s) \cdot...
Let $u_{n} \in \mathcal{L}^{p}, p \geqslant 1$, for all $n \in \mathbb{N}$. What can you say about $u$ and $w$ if you know that \(\lim _{n ightarrow \infty} \int\left|u_{n}-uight|^{p} d...
Do you think the study of organizational ethics should include religious beliefs?
POTI ENTERPRISES LTD. STATEMENT OF INCOME FOR THE YEAR ENDED DECEMBER 31 (current year) SALES $600,000 COST OF SALES: $50,000 OPENING INVENTORY 250,000 PURCHASES 300,000 CLOSING INVENTORY 60,000...
10. Describe a qualified defined contribution plan for the self-employed and discuss the advantages and disadvantages in adopting this type of plan. 11. Describe a SEP IRA and discuss the advantages...
7.) In 1999, the average percentage of women who received prenatal care per country is 80.1%. Table #7.3.9 contains the percentage of woman receiving prenatal care in 2009 for a sample of countries...
Describe A demographic profile of the population and community that will be served through the reinvented Human Service program. The description must include all eligibility requirements (i.e.,...
You work for a major financial institution. Your branch handles customer calls from a wide variety of individuals. Recently, you've noticed an increase in calls from individuals from African...
A gas initially at a subsonic velocity enters an adiabatic converging duct. Discuss how this affects (a) the velocity, (b) the temperature, (c) the pressure, and (d) the density of the fluid.
Using a graphing utility, graph y = cot -1 x.
What was the problem facing Sears? What management, organization, and technology factors contributed to this problem?
What solution did Sears select? What was the role of technology in this solution?
How effective was the solution Sears selected? Explain your answer.
question 6 Timely Inc. produces luxury bags. The budgeted sales and production for the next three months are as follows july. august september Sales, in units 1,115. 1229. 1302 Production. in units...
On May 12 Zimmer Corporation placed in service equipment (seven-year property) with a basis of $220,000. This was Zimmer's only asset acquired during the year. Calculate the maximum depreciation...
Power Manufacturing has equipment that it purchased 7 years ago for $2,550,000. The equipment was used for a project that was intended to last for 9 years and was being depreciated over the life of...

Let ((X, mathscr{A}, mu)) be a finite measure space and let (u in mathcal{L}^{1}(mu)) be strictly positive

Question:

Step by Step Answer:

We remark that y logy is concave Theref...View the full answer

Measures Integrals And Martingales

Students also viewed these Business questions