Step function and simple function

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I am trying to understand clearly whether there is any relation between the set of step function and simple function. Till now I have got that none is the subset of other. The Dirichlet function is a simple function(correct me if i am wrong.) Which is not a step function.
I am not able to conclude clearly. Kindly help me with this.

functions special-functions

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asked Jul 27 at 5:07

Ppp

174

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up vote
0
down vote

favorite

I am trying to understand clearly whether there is any relation between the set of step function and simple function. Till now I have got that none is the subset of other. The Dirichlet function is a simple function(correct me if i am wrong.) Which is not a step function.
I am not able to conclude clearly. Kindly help me with this.

functions special-functions

share|cite|improve this question

asked Jul 27 at 5:07

Ppp

174

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

I am trying to understand clearly whether there is any relation between the set of step function and simple function. Till now I have got that none is the subset of other. The Dirichlet function is a simple function(correct me if i am wrong.) Which is not a step function.
I am not able to conclude clearly. Kindly help me with this.

functions special-functions

share|cite|improve this question

asked Jul 27 at 5:07

Ppp

174

I am trying to understand clearly whether there is any relation between the set of step function and simple function. Till now I have got that none is the subset of other. The Dirichlet function is a simple function(correct me if i am wrong.) Which is not a step function.
I am not able to conclude clearly. Kindly help me with this.

functions special-functions

share|cite|improve this question

asked Jul 27 at 5:07

Ppp

174

share|cite|improve this question

share|cite|improve this question

asked Jul 27 at 5:07

Ppp

174

asked Jul 27 at 5:07

Ppp

174

asked Jul 27 at 5:07

Ppp

174

174

add a comment | 

add a comment | 

2 Answers
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up vote
2
down vote

Step functions are simple (because intervals are measurable), not all simple functions are step functions (because of your example for instance). More conceptually, step functions are also sort of the Riemann analogue of Lebesgue's simple functions, in that Riemann integrals are obtained by integrating a sequence of step functions converging pointwise to the true integrand.

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answered Jul 27 at 5:11

Ian

65k24681

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks for the clarification.
  – Ppp
  Jul 27 at 5:22
  

  
  
  
  

add a comment | 

up vote
1
down vote

A simple function by definition is a function taking only finitely many values. A step function is a finite linear combination of indicator functions of intervals.

In particular, each step function takes finitely many values, hence is a simple function. The other implication does not hold.

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answered Jul 27 at 5:16

Jonas Lenz

326212

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks for the clarification.
  – Ppp
  Jul 27 at 5:22
  

  
  
  
  

add a comment | 

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2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
2
down vote

Step functions are simple (because intervals are measurable), not all simple functions are step functions (because of your example for instance). More conceptually, step functions are also sort of the Riemann analogue of Lebesgue's simple functions, in that Riemann integrals are obtained by integrating a sequence of step functions converging pointwise to the true integrand.

share|cite|improve this answer

answered Jul 27 at 5:11

Ian

65k24681

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks for the clarification.
  – Ppp
  Jul 27 at 5:22
  

  
  
  
  

add a comment | 

up vote
2
down vote

Step functions are simple (because intervals are measurable), not all simple functions are step functions (because of your example for instance). More conceptually, step functions are also sort of the Riemann analogue of Lebesgue's simple functions, in that Riemann integrals are obtained by integrating a sequence of step functions converging pointwise to the true integrand.

share|cite|improve this answer

answered Jul 27 at 5:11

Ian

65k24681

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks for the clarification.
  – Ppp
  Jul 27 at 5:22
  

  
  
  
  

add a comment | 

up vote
2
down vote

up vote
2
down vote

Step functions are simple (because intervals are measurable), not all simple functions are step functions (because of your example for instance). More conceptually, step functions are also sort of the Riemann analogue of Lebesgue's simple functions, in that Riemann integrals are obtained by integrating a sequence of step functions converging pointwise to the true integrand.

share|cite|improve this answer

answered Jul 27 at 5:11

Ian

65k24681

Step functions are simple (because intervals are measurable), not all simple functions are step functions (because of your example for instance). More conceptually, step functions are also sort of the Riemann analogue of Lebesgue's simple functions, in that Riemann integrals are obtained by integrating a sequence of step functions converging pointwise to the true integrand.

share|cite|improve this answer

answered Jul 27 at 5:11

Ian

65k24681

share|cite|improve this answer

share|cite|improve this answer

answered Jul 27 at 5:11

Ian

65k24681

answered Jul 27 at 5:11

Ian

65k24681

answered Jul 27 at 5:11

Ian

65k24681

65k24681

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks for the clarification.
  – Ppp
  Jul 27 at 5:22
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks for the clarification.
  – Ppp
  Jul 27 at 5:22
  

  
  
  
  

Thanks for the clarification.
– Ppp
Jul 27 at 5:22

Thanks for the clarification.
– Ppp
Jul 27 at 5:22

add a comment | 

up vote
1
down vote

A simple function by definition is a function taking only finitely many values. A step function is a finite linear combination of indicator functions of intervals.

In particular, each step function takes finitely many values, hence is a simple function. The other implication does not hold.

share|cite|improve this answer

answered Jul 27 at 5:16

Jonas Lenz

326212

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks for the clarification.
  – Ppp
  Jul 27 at 5:22
  

  
  
  
  

add a comment | 

up vote
1
down vote

A simple function by definition is a function taking only finitely many values. A step function is a finite linear combination of indicator functions of intervals.

In particular, each step function takes finitely many values, hence is a simple function. The other implication does not hold.

share|cite|improve this answer

answered Jul 27 at 5:16

Jonas Lenz

326212

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks for the clarification.
  – Ppp
  Jul 27 at 5:22
  

  
  
  
  

add a comment | 

up vote
1
down vote

up vote
1
down vote

A simple function by definition is a function taking only finitely many values. A step function is a finite linear combination of indicator functions of intervals.

In particular, each step function takes finitely many values, hence is a simple function. The other implication does not hold.

share|cite|improve this answer

answered Jul 27 at 5:16

Jonas Lenz

326212

A simple function by definition is a function taking only finitely many values. A step function is a finite linear combination of indicator functions of intervals.

In particular, each step function takes finitely many values, hence is a simple function. The other implication does not hold.

share|cite|improve this answer

answered Jul 27 at 5:16

Jonas Lenz

326212

share|cite|improve this answer

share|cite|improve this answer

answered Jul 27 at 5:16

Jonas Lenz

326212

answered Jul 27 at 5:16

Jonas Lenz

326212

answered Jul 27 at 5:16

Jonas Lenz

326212

326212

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks for the clarification.
  – Ppp
  Jul 27 at 5:22
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks for the clarification.
  – Ppp
  Jul 27 at 5:22
  

  
  
  
  

Thanks for the clarification.
– Ppp
Jul 27 at 5:22

Thanks for the clarification.
– Ppp
Jul 27 at 5:22

add a comment | 

 

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