Question: Let be a -finite measure and let be a nonnegative -additive and finite function, both defined on the measurable space ((, A). Then

Let μ be a σ-finite measure and let φ be a nonnegative σ-additive and finite function, both defined on the measurable space ((, A). Then show that φ << μ is equivalent to the following: for every ( > 0, there exists ( = ( (() > 0 such that μ (A) < ( implies φ (A) < (.

Step by Step Solution

★★★★★

3.28 Rating (157 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

Suppose that for any 0 there exists 0 such that A implies A Then if A 0 it follows t... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

742-M-S-P (6863).docx

120 KBs Word File

Students Have Also Explored These Related Statistics Questions!

If X is a r.v. defined on the measurable space ((, A), then |X| is also a r.v. By an example, show that the converse need not be true.

Given the finite measures n, n = 1, 2,...., on the measurable space ((, A), define a probability measure such that n

Let f be a nonnegative continuous function on [0,1]. (a) Show that (b) Use part (a) to evaluate xf(sin x) dx = (n/2) | f(sin x) dx. x sin x cos x ax.

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...

ttth Suppose that the sequence of bags {Bn | n N} is recursively enumerated by the computable function e(n, x) = fn(x), [7 marks] Hence prove that the set of all recursive bags cannot be recursively...

1 2.3 Definition of a Discrete Probability Function Definition: Let S be a discrete sample space from some experiment. A function P, defined on all events in S, is said to be a probability function...

Please explain to me clearly and show how to do the two suppositions in the (question) file. Please take a look at the (notes) file and see the material up to the questions will be needed to answer...

Is the set function $\gamma$ of Example 4.5 (ii) still a measure on the measurable space $(\mathbb{R}, \mathscr{B}(\mathbb{R}))$ ? Is it still a measure on the measurable space \((\mathbb{Q},...

Applied Mathematics and Computation 95 (1998) 181192 Love dynamics: The case of linear couples Sergio Rinaldi 1 Centro Teoria dei Sistemi, CNR, Politecnico di Milano, Via Ponzio 34/5, 20133 Milan,...

ANSI-SPARC6 Programming Language Compilation Write notes on each of the following topics: (a) the implementation of labels and jumps in a recursive, block structured programming language [7 marks]...

The Durham Inn, a proposed 80-room limited service inn, will cost $4,860,000 to build. The owners want a 14 percent return on investment after their company pays income taxes of 22 percent. Annual...

Review each of the following situations independently: Upon learning that the replacement cost of its Caterpillar tractor was Shs.100,000 more than its book value, Malta Company added Shs.100,000 to...

What kinds of languages are regular expressions unable to recognize? [ 2 marks ]

You are a PMP, and PMI has contacted you regarding an investigation of your best friend who is also a PMP at your company. How should you proceed? A. Cooperate fully with PMI. B. Protest that this is...

An oil company purchased an option on land in Alaska. Preliminary geologic studies assigned the following prior probabilities. P(high-quality oil) = 0.50 P(medium-quality oil) = 0.20 P(no oil) = 0.30...

Small cars get better gas mileage, but they are not as safe as bigger cars. Small cars accounted for 18% of the vehicles on the road, but accidents involving small cars led to 11,898 fatalities...

(DuPont analysis) Triangular Chemicals has total assets of $93 million, a return on equity of 44 percent, a net profit margin of 5.4 percent, and an equity multiplier of 2.33. How much are the firm's...

Create an enumeration named Planet that holds the names for the eight planets in our solar system, starting with MERCURY equal to 1 and ending with NEPTUNE. Write a program named Planets that prompts...

5. Interest Rate Risk Collapse . Both Bonds A and B are priced at par since their YTMs are the same as their coupon rate of 7%. What happens to their prices if interest rate in the market rises by...