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The set $D_v= xin mathbbR^N: D_vf(x) text exists $ is measurable.

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$begingroup$

Let $f:mathbbR^nto mathbbR$ be Lipsticz function. The directional derivative in the direction of $mathbfv$ is given by $$ D_mathbfvf(mathbfx)=lim_tto 0dfracf(mathbfx+tmathbfv)-f(mathbfx)t. $$ Define a set $$ D_mathbfv= mathbfxin mathbbR^n:D_mathbfvf(mathbfx)text exists .$$ Show that $D_mathbfv$ is measurable.

Please help me in proving that.

measure-theory

share|cite|improve this question

asked yesterday

XYZABCXYZABC

344110

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What have you tried? What do you know? Do you know that $limsup_tto 0 g_t(x)$ of measurable functions for each $t$ is measurable?
  $endgroup$
  – Shashi
  yesterday
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes that I know. But how does this help could you please give me some more hint. Then probably I can proceed.
  $endgroup$
  – XYZABC
  yesterday
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Then the set you are looking for is $x : limsup_tto 0 g_t(x)-liminf_tto 0g_t(x)=0 $ for a suitable chosen $g_t(x)$... Do you get it?
  $endgroup$
  – Shashi
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Okay, I got it. Thanks.
  $endgroup$
  – XYZABC
  yesterday
  

  
  
  
  

add a comment | 

0

$begingroup$

Let $f:mathbbR^nto mathbbR$ be Lipsticz function. The directional derivative in the direction of $mathbfv$ is given by $$ D_mathbfvf(mathbfx)=lim_tto 0dfracf(mathbfx+tmathbfv)-f(mathbfx)t. $$ Define a set $$ D_mathbfv= mathbfxin mathbbR^n:D_mathbfvf(mathbfx)text exists .$$ Show that $D_mathbfv$ is measurable.

Please help me in proving that.

measure-theory

share|cite|improve this question

asked yesterday

XYZABCXYZABC

344110

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What have you tried? What do you know? Do you know that $limsup_tto 0 g_t(x)$ of measurable functions for each $t$ is measurable?
  $endgroup$
  – Shashi
  yesterday
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes that I know. But how does this help could you please give me some more hint. Then probably I can proceed.
  $endgroup$
  – XYZABC
  yesterday
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Then the set you are looking for is $x : limsup_tto 0 g_t(x)-liminf_tto 0g_t(x)=0 $ for a suitable chosen $g_t(x)$... Do you get it?
  $endgroup$
  – Shashi
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Okay, I got it. Thanks.
  $endgroup$
  – XYZABC
  yesterday
  

  
  
  
  

add a comment | 

$begingroup$

Let $f:mathbbR^nto mathbbR$ be Lipsticz function. The directional derivative in the direction of $mathbfv$ is given by $$ D_mathbfvf(mathbfx)=lim_tto 0dfracf(mathbfx+tmathbfv)-f(mathbfx)t. $$ Define a set $$ D_mathbfv= mathbfxin mathbbR^n:D_mathbfvf(mathbfx)text exists .$$ Show that $D_mathbfv$ is measurable.

Please help me in proving that.

measure-theory

share|cite|improve this question

asked yesterday

XYZABCXYZABC

344110

$endgroup$

share|cite|improve this question

asked yesterday

XYZABCXYZABC

344110

share|cite|improve this question

asked yesterday

XYZABCXYZABC

344110

asked yesterday

XYZABCXYZABC

344110

asked yesterday

XYZABCXYZABC

344110