Let ((X, mathscr{A}, mu)) be a (sigma)-finite measure space and (mathscr{G} subset mathscr{A}) be a sub- (sigma)-algebra.

Question:

Let $(X, \mathscr{A}, \mu)$ be a $\sigma$-finite measure space and $\mathscr{G} \subset \mathscr{A}$ be a sub- $\sigma$-algebra. Show that, in general,

\[\int \mathbb{E}^{\mathscr{G}} u d \mu \leqslant \int u d \mu, \quad u \in L^{1}(\mathscr{A}), u \geqslant 0\]

with equality holding only if $\left.\muight|_{\mathscr{G}}$ is $\sigma$-finite.

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

Step by Step Answer:

Related Book For answer-question

Measures Integrals And Martingales

ISBN: 9781316620243

2nd Edition

Authors: René L. Schilling

See More Books

Question Posted: Jan 25, 2024 01:07 AM

See More Questions

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...
The devil's staircase. Recall the construction of Cantor's ternary set from Problem 7.12. Denote by $I_{n}^{1}, \ldots, I_{n}^{n}-1$ the intervals which make up $[0,1] \backslash C_{n}$ arranged...
How does unemployment behave over the business cycle?
Bullets of mass 3.0g are fired in parallel paths with speeds of 220m/s through a hole 3.0mm in diameter, how far from the hole must you be to detect a 1.0-cm-diameter spread in the beam of bullets?
The world of shapes is much richer than the shapes included in the inheritance hierarchy of Fig. 11.3. Write down all the shapes you can think ofboth two dimensional and three-dimensionaland form...
Refer to the information from QS 21-18. Compute the variable overhead spending variance and the variable overhead efficiency variance and classify each as favorable or unfavorable. Data From QS 21-18...
The adjusted trial balance of Big Ritas Music Company at June 30, 2016, follows: Requirements 1. Prepare Big Ritas multi-step income statement for the year ended June 30, 2016. 2. Journalize Big...
Ports should also be able to adapt to global and regional political or social changes. Historically, it is apparent that unexpected sociopolitical developments have demanded ports to change their...
Prove Corollaries 27.14 and 27.15 . Data from corollary 27.14 Data from corollary 27.15 Corollary 27.14 (conditional Fatou's lemma) Let (X,A,) be a o-finite measure space, CA be a sub-o-algebra, such...
Consider on the measure space $\left([0, \infty), \mathscr{B}[0, \infty),\left.\lambda^{1}ight|_{[0, \infty)}ight)$ the filtration defined by \(\mathscr{A}_{n}:=\sigma([0,1),[0,2), \ldots,[n-1,...
Design a frequency divider circuit to output a 100-kHz signal. The input frequency is 5 MHz. Simulate (functional) the counter. (a) Use LPM_COUNTER. (b) Use an HDL.
In the "Auditor's Guide to Information Systems Auditing", A subset of the described IT definitions is listed below. Given what was discussed and your research, place a risk designation of high,...
E-portfolios are different from a resume, which is normally tailored to the position, I see portfolios as a full representation of the student that has minimal need for change, i.e., it is designed...
When there is an export and import control barrier, which of the strategies is not appropriate in this situation?
Which accounts may be set up to use account reconciliation - a. all asset, liability and equity accounts b. only asset and equity or capital accounts c. only bank and credit card asset accounts d....
There is a very useful generation fonction that related to the Legendre polynomials:
"Steve, who is single, has $100,000 of salary, $10,000 of income from a limited partnership, and a $25,000 passive loss from a real estate rental activity in which he actively participates. His...
Perform the indicated operations. In designing a cam for a fire engine pump, the expression is used. Simplify this expression. (3) (3 4 32
The Graduate Record Examination (GRE) subject test in economics was a multiple-choice measure of knowledge and analytical ability in economics that was used mainly as an entrance criterion for...
Lets get some more experience with the six steps in applied regression. Suppose that youre interested in buying an Apple iPod (either new or used) on eBay (the auction website) but you want to avoid...
Carlos inherits 100 shares of Allied Corporation stock from his father. The stock cost his father $8,000 and had a $25,000 FMV on the date of his fathers death in 2017. The alternate valuation date...
CPA Service Company received $200,000 cash from partners (owners), paid off accounts payable of $65,000, and paid auditor salaries of $105,000. What net effect do these effect have on shareholders'...
BuyCo, Incorporated, holds 25 percent of the outstanding shares of Marqueen Company and appropriately applies the equity method of accounting. Excess cost amortization (related to a patent)...
You need to prepare 100.0 mL of a pH 4.00 buffer solution using 0.100 M benzoic acid (pK = 4.20) and 0.180 M sodium benzoate. How many milliliters of each solution should be mixed to prepare this...

Let ((X, mathscr{A}, mu)) be a (sigma)-finite measure space and (mathscr{G} subset mathscr{A}) be a sub- (sigma)-algebra.

Question:

Step by Step Answer:

For this problem it is helpful to distinguis...View the full answer

Measures Integrals And Martingales

Students also viewed these Business questions