Question: Let (X, A, ) be a measure space and let (fn)neN, (gn)neN, and f, g be extended real-valued A-measurable functions on DE A. Suppose

Let (X, A, ) be a measure space and let (fn)neN, (gn)neN,

Let (X, A, ) be a measure space and let (fn)neN, (gn)neN, and f, g be extended real-valued A-measurable functions on DE A. Suppose that 1. limno fn = f and limn-Inga.e. on D. 2. (gn) and g are all integrable on D and limno fp9nd = SD gd. 3. fn9n on D for every n E N. Prove that f is integrable on D and limno fp fndu = fpfdu.

Step by Step Solution

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

The detailed ... 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

Students Have Also Explored These Related Mathematics Questions!

Suppose that fn f and gn g, as n , uniformly on some set E R. a) Prove that fn + gn f + g and fn f f, as n uniformly on E for all R. b) Prove that fngn fg pointwise on E. c) Prove that if...

For n 1, let gn and g be functions defined on E ( Rk into R, and recall that {gn} is said to converge continuously to g on E, if for every x ( E, gn(xn) g(x) whenever xn x, as n. Then show that if...

Let fn be integrable on [0, 1] and fn f uniformly on [0, 1]. Show that if bn f 1 as n , then limo, fn(x) dx= 10 f(x) dx.

Egorov's theorem. Let $(X, \mathscr{A}, \mu)$ be a finite measure space and $f_{n}: X ightarrow \mathbb{R}, n \in \mathbb{N}$, a sequence of measurable functions. Prove the following assertions....

Borel-Cantelli lemma (1) - the direct half. Prove the following theorem. Theorem (Borel-Cantelli lemma). Let $(\Omega, \mathscr{A}, \mathbb{P})$ be a probability space. For every sequence...

Extend Proposition 4.3 ( i ) 4.3 ( i ) , (iv) and (v) to finitely many sets A 1 , A 2 , , A N A A 1 , A 2 , , A N A . Data from proposition 4.3 Let (X, A,) be a measure space and A, B, An, BnA,...

(i) Show that $\lambda^{1}((a, b))=b-a$ for all $a, b \in \mathbb{R}, a \leqslant b$. [ approximate $(a, b)$ by half-open intervals and use Proposition 4.3 (vi), (vii).] (ii) Let \(H \subset...

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...

Please watch the video link and answer the following questions. https://youtu.be/VtvjbmoDx-I On January 22,1984, Apple produced one of the most famous Super Bowl ads of all time. Directed by Ridley...

Carpenter Inc. had a balance of $98,000in its quality-assurancewarranty liability account as of December 31, 2023. In 2024,Carpenter's warranty expenditures paid were $463,000. Its warrantyexpense 2...

An automobile accident causes both the driver and passenger front airbags to deploy. (a) If the vehicle was traveling at a speed of 88.6 km/h and is now at rest, find the change in momentum for both...

JST Services employs 50 workers who are each paid $900 per week. JST allows each employee two weeks of paid vacation per year. In addition, the company allows each employee one week of paid sick...

The budgeting process does not involve which of the following activities? a. Execution of plans to achieve goals. b. Specific goals are established. c. Periodic comparison of actual results to goals....

Consider the following code segment: int numFeet, numInches; //number of feet // number of inches int totalInches -52: //total number of inches Write the statements to convert the total number of...

Identify which question you are answering. Round all decimals. On January 1, Year 4 A purchased 80% of the outstanding common shares of B. The acquisition differential of $20,000 was allocated as...

Give me a conducted feasibility study for a mechanical engineering consultancy firm?

The level curve of the function h(x, y) = 16 a2 - ? that passes through the point (0, 2, h(0, 2)) is given by the set: %3D In your complete answer, sketch the graph of the selected level curve. OC =...

You are analyzing the following information Company Name Price Earnings Ratio Dividend Yield Market Price Frankfurter Inc 18 1.5% 200 Eggo Corp 150 0 400 Dialogic ltd 6 5. % 50 What is Dialogic's...

Let (X, A, ) be a measure space and let (fn)neN, and f, g be extended real-valued A-measurable and integrable functions on DE A. Suppose that 1. limn- fn = fa.e. on D. 2. limno fp fndu = fpfdu. 3....

Let (fn) be a sequence of extended real-valued measurable functions on X and let f be an extended real-valued function which is finite a.e. on X. Suppose limn- fn = fa.e. on X. Let a = [0, (X)) be...

ENSURE WORK IS ORIGINAL OR I REJECT AND REPORT YOU TON THE SUPPORT INSTANTLYThe end result does now not should be taken care of. You may additionally expect n 6 length(xs). [Hint: Function least is a...

Make sense of the thoughts of static and dynamic chains, giving an illustration of a circumstance in which they contrast (for example reference to some unacceptable one would get to the wrong...

Let A, B be sets. Define: (a) the Cartesian product (A B) (b) the set of relations R between A and B (c) the identity relation A on the set A [3 marks] Suppose S, T are relations between A and B, and...

Microkernel operating systems aim to address perceived modularity and reliability issues in traditional "monolithic" operating systems. (i) Describe the typical architecture of a microkernel...