I’m preparing for my qualifying exam and i want some imput in my solution of this problem.

Let $f:Bbb{R} to Bbb{R}$ measurable and $1-$periodic(i.e $f(x+1)=f(x),forall x in Bbb{R}$)

If $int_0^1|f(x+t)-f(x)|dx leq 1,forall t in Bbb{R}$,prove that $f in L^1[0,1]$

My proof goes like this:

we have that $$int_0^1 int_0^1 |f(x+t)-f(x)|dxdt leq 1 Longrightarrow int_0^1 int_0^1 |f(x+t)-f(x)|dtdx leq 1$$

By Fubini: we have that the function $g_x(t)=|f(x+t)-f(x)|$ is integrable on $ [0,1]$ for almost every $x in [0,1]$ Let $A$ be the set of such $x$. By periodicity we have that $$int_I g_x(t)dx=int_J g_x(t)dx$$ for every interval $J,I$ with length $1$ for all $x in A$

Now $$1 geq int_A int_0^1 |f(x+t)-f(x)|dtdx=$$ $$int_A int_{-x}^{-x+1} |f(x+t)-f(x)|dtdx=int_A int_0^1 |f(z)-f(x)|dxdz$$

Since $A^c$ has measure zero we conclude the $G(x,z)=f(x)-f(z) in L^1([0,1]^2)$ so again by Fubini it is easy to conclude that $f in L^1[0,1]$

Is there a flaw in my argument?

Thnak you in advance

Mathematics Asked by Marios Gretsas on January 1, 2021

0 Answers1 Asked on November 24, 2021 by zest

abstract algebra group homomorphism homological algebra quotient group tensor products

0 Asked on November 24, 2021 by silkking

1 Asked on November 24, 2021

calculus integration ordinary differential equations real analysis

1 Asked on November 24, 2021 by mathnewbie

4 Asked on November 24, 2021

eigenvalues eigenvectors linear algebra linear transformations operator algebras

1 Asked on November 21, 2021

2 Asked on November 21, 2021 by user686123

2 Asked on November 21, 2021

1 Asked on November 21, 2021 by threnody

1 Asked on November 21, 2021 by user785957

bessel functions hypergeometric function special functions summation summation method

2 Asked on November 21, 2021

2 Asked on November 21, 2021 by shaheer-ziya

0 Asked on November 21, 2021 by user297564

1 Asked on November 21, 2021 by damianodamiano

exponential distribution integration probability probability distributions

0 Asked on November 21, 2021 by physicslab1

0 Asked on November 21, 2021

0 Asked on November 21, 2021

boundary value problem laplacian linear pde mathematical physics partial differential equations

3 Asked on November 21, 2021 by shubhrajit-bhattacharya

alternative proof combinatorics number theory polynomials prime numbers

1 Asked on November 21, 2021 by isato

Get help from others!

Recent Questions

- MouseLook Script “Pops” back to the last value when the script is enabled after being disabled or destroyed
- Unity app crashes when using unmodified custom Android manifest (didn’t find class “UnityPlayerActivity”)
- How do i draw a ray in unity
- How to test consistency of responses?
- How can I understand these variograms?

Recent Answers

- Philipp on How do i draw a ray in unity
- Justin Markwell on Unity app crashes when using unmodified custom Android manifest (didn’t find class “UnityPlayerActivity”)
- kjetil b halvorsen on How to test consistency of responses?
- eric_kernfeld on How to test consistency of responses?
- DMGregory on MouseLook Script “Pops” back to the last value when the script is enabled after being disabled or destroyed

© 2022 AnswerBun.com. All rights reserved.