Dilations. Mimic the proof of Theorem 5.8 (i) and show that (t cdot B:={t b: b in

Question:

Dilations. Mimic the proof of Theorem 5.8 (i) and show that $t \cdot B:=\{t b: b \in B\}$ is a Borel set for all $B \in \mathscr{B}\left(\mathbb{R}^{n}ight)$ and $t>0$. Moreover,

\[\lambda^{n}(t \cdot B)=t^{n} \lambda^{n}(B) \quad \forall B \in \mathscr{B}\left(\mathbb{R}^{n}ight), \forall t>0\]

Data from theorem 5.8

$the translation invariance of u and \", we see that (1)= k(1)([0,7)"), X" (1)= k(1) X" ([0,7)"), ([0, 1)") =$

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 22, 2024 03:05 AM

See More 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...
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...
Completing a detail in the proof of Theorem 12.5, let G be a finite group consisting of the identity isometry and rotations about one point P in the plane. Show that G is cyclic, generated by the...
Dew Company employs standard costing and applies overhead based upon machine labor hours. Monthly production is scheduled at 500 units. Standard and actual data for the month is as follows: Standards...
Determine the moments of inertia I x and I y of the area shown with respect to centroidal axes respectively parallel and perpendicular to side AB. B. 1.1 in. 0.7 in. 3.6 in.
Use the information provided to estimate how much of the unemployment in Ames and El Centro in 2009 was cyclical and how much was natural. Explain your assumptions in arriving at your estimates. The...
During the last football season, the Brutes managed to gain an average of 203 yards per game while their opponents averaged 288 yards per game. Fire the coach?
Hinshaw Company purchased a new machine on October 1, 2014, at a cost of $90,000. The company estimated that the machine has a salvage value of $8,000. The machine is expected to be used for 70,000...
Eliot Sprinkler Systems produces equipment for lawn irrigation. One of the parts used in selected Eliot equipment is a specialty nozzle. The budgeting team is now determining the purchase...
Invariant measures. Let $(X, \mathscr{A}, \mu)$ be a finite measure space where $\mathscr{A}=\sigma(\mathscr{G})$ for some $\cap$-stable generator $\mathscr{G}$. Assume that \(\theta: X...
Show that the half-open intervals $\mathscr{J}$ in $\mathbb{R}^{n}$ are stable under finite intersections. [ check that \(\left.\mathrm{X}_{i=1}^{n}\left[a_{i}, b_{i}ight) \cap...
Find the limit (if it exists). If the limit does not exist, explain why. x + y lim (x, y) (0,0) x + y
Lacoste t-shirts come with an average price of $ 120 a piece, at their factory outlet with a std. deviation of $ 17. But at the Seasonal Sale (Discount) outlets of these t- shirts, it was also...
The graph above is the U.S. market for some imported good. Supply is a flat curve. The U.S. can import the Chinese good for $40 and the Mexican good for $48. Assume the U.S. imposes $10 tariffs on...
What is the formula to find total dividend and payout ratio? This is the information I have: the amount of shares the company holds and the last dividend paid. Lastly, will there be enough cash to...
Prepare an Orientation Plan for the select position utilizing the theories discussed in the text. Discuss appropriate and effective orientation plans and your decision-making process for selecting a...
Simplify the expression 2x + (x + 1) into a single x + 1 fraction. The numerator of your answer is: The denominator of your answer is:
Cheap Travel borrowed $60,000 on October 1 by signing a note payable to Community Bank. The interest expense for each month is $250. The loan agreement requires Cheap to pay interest on December 31....
What are some of the possible sources of information about a company that could be used for determining the companys competitive stance?
Let $\tau$ be a stopping time. Show that $\mathbb{1}_{[0, \tau)}$ is $\mathscr{P}$ measurable.
Let \(T
Show that every right or left continuous adapted process is $\mathscr{P}$ measurable.
Given the solution matrix on the calculator with columns x, y, and z. Determine the type of solution the original problem would have. 204 5 0 3 27
What is the major difference between a traditional society and a society of organizations? What are the three reasons time is likely the most powerful asset you have? What are the three examples of...
Look at the following job requirement for the Oklahoma Bureau of Investigation and the Florida Department of Law Enforcement special agents. Discuss the differences in education between the two...

Dilations. Mimic the proof of Theorem 5.8 (i) and show that (t cdot B:={t b: b in

Question:

(i) The n-dimensional Lebesgue measure X" is invariant under translations, i.e. X" (x+B)=X" (B) VxER", VBEB(R"). (5.3) (ii) Every measure u on (R", B(R")) which is invariant under translations and satisfies K = ([0, 1)")

Step by Step Answer:

First we must make sure that 1B is a Borel set if BE B We consider first rectangles I ab where a b R ...View the full answer

Measures Integrals And Martingales

Students also viewed these Business questions