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Question about lim inf or lim sup

0

$begingroup$

When taking the course of real analysis, the professor wrote the following:

Let $f_n geq 0$ for all $n$.

Let $$a = lim inf int_X f_n dmu + epsilon$$
I think here $epsilon$ is any positive number.
Now, by the definition of $lim inf$, we have
$$forall N, exists, n >N, text such that int_X f_ndmu <a$$

---

My question is, if $f_n$ is decreasing, i.e., $$f_1geq f_2 geq f_3 cdots geq f_n geq cdots,$$ then how could the conclusion hold? (if choosing $epsilon rightarrow 0$)

real-analysis limsup-and-liminf

share|cite|improve this question

asked Mar 27 at 3:18

sleeve chensleeve chen

3,19342256

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Did you mean $exists N forall n>N$?
  $endgroup$
  – Reveillark
  Mar 27 at 3:38
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Reveillark No. But I am not sure if this is a typo made from that professor.
  $endgroup$
  – sleeve chen
  Mar 27 at 4:38
  

  
  
  
  

add a comment | 

0

$begingroup$

When taking the course of real analysis, the professor wrote the following:

Let $f_n geq 0$ for all $n$.

Let $$a = lim inf int_X f_n dmu + epsilon$$
I think here $epsilon$ is any positive number.
Now, by the definition of $lim inf$, we have
$$forall N, exists, n >N, text such that int_X f_ndmu <a$$

---

My question is, if $f_n$ is decreasing, i.e., $$f_1geq f_2 geq f_3 cdots geq f_n geq cdots,$$ then how could the conclusion hold? (if choosing $epsilon rightarrow 0$)

real-analysis limsup-and-liminf

share|cite|improve this question

asked Mar 27 at 3:18

sleeve chensleeve chen

3,19342256

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Did you mean $exists N forall n>N$?
  $endgroup$
  – Reveillark
  Mar 27 at 3:38
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Reveillark No. But I am not sure if this is a typo made from that professor.
  $endgroup$
  – sleeve chen
  Mar 27 at 4:38
  

  
  
  
  

add a comment | 

$begingroup$

When taking the course of real analysis, the professor wrote the following:

Let $f_n geq 0$ for all $n$.

Let $$a = lim inf int_X f_n dmu + epsilon$$
I think here $epsilon$ is any positive number.
Now, by the definition of $lim inf$, we have
$$forall N, exists, n >N, text such that int_X f_ndmu <a$$

---

My question is, if $f_n$ is decreasing, i.e., $$f_1geq f_2 geq f_3 cdots geq f_n geq cdots,$$ then how could the conclusion hold? (if choosing $epsilon rightarrow 0$)

real-analysis limsup-and-liminf

share|cite|improve this question

asked Mar 27 at 3:18

sleeve chensleeve chen

3,19342256

$endgroup$

share|cite|improve this question

asked Mar 27 at 3:18

sleeve chensleeve chen

3,19342256

share|cite|improve this question

asked Mar 27 at 3:18

sleeve chensleeve chen

3,19342256

asked Mar 27 at 3:18

sleeve chensleeve chen

3,19342256

asked Mar 27 at 3:18

sleeve chensleeve chen

3,19342256