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A concrete question in Real Analysis

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$begingroup$

Suppose that $x^k$ is a set of $n$-dimensional vectors. $omegainOmega$ and $Omega$ is a measurable set. Denote $v_i$ as the $i$-th component of $v$ if $v$ is a vector. $rho(cdot)$ is a positive function on $a.e. Omega$, $M(cdot)$ and $q(cdot)$ are $ntimes n$ matrix and $n$-dimensional vector on $omega$, respectively. Moreover, let
$$J:=j: varlimsuplimits_k x^k_j=+infty$$
and $x^k_i$ is bounded below for each $iin1,2,cdots,n$.
If we further have
$$varlimsuplimits_k|int_Omega_0x^k_i[M(omega)x^k+q(omega)]_irho(omega)domega|<+infty$$
for any subset $Omega_0subsetOmega$ and $iin J$. Besides, we know that $[M(omega)x^k+q(omega)]_i$ is bounded for any $iin1,2,cdots,n$. Then, could we claim that
$$limlimits_k [M(omega)x^k+q(omega)]_i=0quad a.e. omegainOmega.$$
for any $iin J$?

real-analysis limits lebesgue-integral

share|cite|improve this question

asked Mar 21 at 14:04

shmilyshmily

1

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  You are more likely to get responses if you provide context and elaborate on your own attempts on solving the problem.
  $endgroup$
  – PierreCarre
  Mar 21 at 14:16
  

  
  
  
  

add a comment | 

0

$begingroup$

Suppose that $x^k$ is a set of $n$-dimensional vectors. $omegainOmega$ and $Omega$ is a measurable set. Denote $v_i$ as the $i$-th component of $v$ if $v$ is a vector. $rho(cdot)$ is a positive function on $a.e. Omega$, $M(cdot)$ and $q(cdot)$ are $ntimes n$ matrix and $n$-dimensional vector on $omega$, respectively. Moreover, let
$$J:=j: varlimsuplimits_k x^k_j=+infty$$
and $x^k_i$ is bounded below for each $iin1,2,cdots,n$.
If we further have
$$varlimsuplimits_k|int_Omega_0x^k_i[M(omega)x^k+q(omega)]_irho(omega)domega|<+infty$$
for any subset $Omega_0subsetOmega$ and $iin J$. Besides, we know that $[M(omega)x^k+q(omega)]_i$ is bounded for any $iin1,2,cdots,n$. Then, could we claim that
$$limlimits_k [M(omega)x^k+q(omega)]_i=0quad a.e. omegainOmega.$$
for any $iin J$?

real-analysis limits lebesgue-integral

share|cite|improve this question

asked Mar 21 at 14:04

shmilyshmily

1

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  You are more likely to get responses if you provide context and elaborate on your own attempts on solving the problem.
  $endgroup$
  – PierreCarre
  Mar 21 at 14:16
  

  
  
  
  

add a comment | 

$begingroup$

Suppose that $x^k$ is a set of $n$-dimensional vectors. $omegainOmega$ and $Omega$ is a measurable set. Denote $v_i$ as the $i$-th component of $v$ if $v$ is a vector. $rho(cdot)$ is a positive function on $a.e. Omega$, $M(cdot)$ and $q(cdot)$ are $ntimes n$ matrix and $n$-dimensional vector on $omega$, respectively. Moreover, let
$$J:=j: varlimsuplimits_k x^k_j=+infty$$
and $x^k_i$ is bounded below for each $iin1,2,cdots,n$.
If we further have
$$varlimsuplimits_k|int_Omega_0x^k_i[M(omega)x^k+q(omega)]_irho(omega)domega|<+infty$$
for any subset $Omega_0subsetOmega$ and $iin J$. Besides, we know that $[M(omega)x^k+q(omega)]_i$ is bounded for any $iin1,2,cdots,n$. Then, could we claim that
$$limlimits_k [M(omega)x^k+q(omega)]_i=0quad a.e. omegainOmega.$$
for any $iin J$?

real-analysis limits lebesgue-integral

share|cite|improve this question

asked Mar 21 at 14:04

shmilyshmily

1

$endgroup$

share|cite|improve this question

asked Mar 21 at 14:04

shmilyshmily

1

share|cite|improve this question

asked Mar 21 at 14:04

shmilyshmily

1

asked Mar 21 at 14:04

shmilyshmily

1

asked Mar 21 at 14:04

shmilyshmily

1