Question: Let (X, A, ) be a finite measure space. Let fn, f L(X) for all n N such that limn- fn fa.e. on X

Let (X, A, ) be a finite measure space. Let fn, f

Let (X, A, ) be a finite measure space. Let fn, f L(X) for all n N such that limn- fn fa.e. on X and ||fn||2 < M for all n N. Show that limno || fn-f||1 = 0. =

Step by Step Solution

★★★★★

3.40 Rating (159 Votes )

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!

Scenario 2: The CFO of a company is looking at a forecast of a new product launch. Table 1, "Option 1: Product Launch", shows the forecasted undiscounted cash flows of this product launch under a...

Cascade Company was started on January 1, Year 1, when it acquired $165,000 cash from the owners. During Year 1, the company earned cash revenues of $95,300 and incurred cash expenses of $60,200. The...

Mary Walker, president of Rusco Company, considers $46,000 to be the minimum cash balance for operating purposes. As can be seen from the following statements, only $41,000 in cash was available at...

For n ( 1, let fn be real-valued measurable functions defined on (, and suppose that |fn (x)| ( M ( for x ( (. Then for any sequence {xn} with it holds ( 0 a.e. [(], where ( is the Lebesgue measure? ...

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....

Suppose X1, . .., An and Y1, ..., In are random variables on a common probability space such that Var(Xi) = Var(Yi) = 4 for all i; and if i = j p(Xi, Yj) = if i * j (1) O else. Let W = Em Xi and Z =...

2. (Time scaling) A stock price S is governed by the model In S(k + 1) = In 5(k)+w(k) where the period length is 1 month. Let v = E[w(k)] and var[w(k)] for all k. Now suppose the basic period length...

Nancy Shrills is considering an expansion of one of its existing buildings to add more manufacturing space for its kid - friendly noisemakers. Several possible scenarios exist for future cash flows,...

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...

Completion (4). Inner measure and outer measure. Let $(X, \mathscr{A}, \mu)$ be a finite measure space. Define for every $E \subset X$ the outer resp. inner measure \[\begin{aligned} &...

Let $(X, \mathscr{A}, \mu)$ be a finite measure space. Show that (i) $u \in \mathcal{L}^{1}(\mu) \Longleftrightarrow u \in \mathcal{M}(\mathscr{A})$ and \(\sum_{n=0}^{\infty} \mu\{|u| \geqslant...

Can a firms transnational strategy be a source of sustained competitive advantage?

Using the information in problem 14, if Tracys standard deduction is $ 5,950 and her exemption is $ 3,800, what is her taxable income?

Find f ' ( x ) if f ( x ) = l n ( | 4 x | ) . Solution Since f ( x ) = { l n ( 4 x ) i f x > 0 l n ( - 4 x ) i f x 0 it follows that f ' ( x ) = { ( 1 ) * d d x ( 4 x ) = i f x > 0 ( 1 ) * d d x ( -...

What is the standard of care of Doctors. Please reference the authorities.

2. As shown in the figure below a turn-style antenna consists of two dipole antennas fed 90 out of phase with each other and are oriented perpendicular to each other in the same plane. (a) Show that...

From the following particulars calculate variable overhead expenditure variance: Actual Variable Overhead expense: $14,250 Budgeted Variable Overhead expense: $17,000 Actual level of Production: 8000...

Q2) A municipal wastewater treatment plant has an average flow of 9.92 MLD, and 2 horizontal-velocity grit chambers, one chamber will operate as a standby. The peak flow is 2.51 times the average...

3. A 3-o 4-wire ABC system with the line voltage VBC = 294.2/0 V has a Y-connected unbalanced load of ZA = 15/0 N, ZB = 15/0 N and Zc = 10/ 30. Obtain the following: (a) all the line currents. (b)...

A random sample of 30 long distance runners aged 20-25 was selected from a running club. The resting heart rates (in beats per minute) of the runners are shown below. Find the 95% confidence interval...

Let (X, A,p) be a complete measure space and let f and g be nonnegative A-measurable functions on X such that fg e l'(X). For any > 0 let Ex = {r X g(x) > 1}. Show that the function F(X) = Sex...

Consider the measure space ( X , , ) , ( X ) . Suppose that f , f n , g , g n : X R , n N are measurable , i.e, a R : f 1 ( { a } ) ,where isa algebra . Prove that a) ( { x X f ( x ) > n } ) 0 , n ....

I need help with Exercise #3 pleas use the attached materials only to solve the problem. thank you Section 3B Limits of Integrals & Integrals of Limits 93 Case 1: Suppose y (X) 0. By 3.28, there...

Exercise 2.3 Let ($2. E, u) be a positive finite measure space. Show that L, ($2, u()* can be identified with Lo($2, #). (This completes the proof of Theorem 2.13.) The following proves the theorem,...

.Let (X,M, ) be a finite measure space. Suppose X admits a sequence of finite partitions Fn consisting of measurable sets, such that sup{(F) : F Fn} 0 as n . Prove that for any t [0, (X)], there...

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...