Listing some properties of a Borel Measure

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Finally I decided to sign up to the forum which I've always found really helpful! I'm James and I'm a Math student (Master degree). Today I came across this exercise that I can't solve and I was wondering if any of you had some ideas about it. Basically I have:

$ mathbbQ = q_n_n in mathbbN$ where ($q_n ne q_m$ if $n ne m$) and let

$$ mu = sum_n in mathbbN frac1n^2 delta_q_n$$

Where $delta_x$ is Dirac delta centered in $x in mathbbR$.

Let $f : mathbbR rightarrow mathbbR$ such that
if $x gt 0$
$$f(x) = mu ((0,x)) $$
if $x le 0$
$$f(x) = mu ((x,0])$$

Summarize some $f$ properties (in particular, sign, continuity, right/left continuity, total variation, boundedness, monotony, values of $f(0^+)$, $f(0^-)$..) and some $mu$ properties (in particular (absolute continuity, discrete, singular.. respect to the Lebesgue measure). Thank you in advance!

real-analysis measure-theory dirac-delta

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edited Jul 30 at 15:06

asked Jul 30 at 14:53

James Arten

186

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I don't understand your notation. For me $(0,x)$ is a point in $Bbb R^2$ and $mu ((0,x))$ to me is meaningless.
  – DanielWainfleet
  Jul 30 at 15:35
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  We're in $mathbbR$ so it's supposed to be the Measure of the interval $(0,x)$ for $x gt 0$. Is that right?
  – James Arten
  Jul 30 at 15:38
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  You'll find that simple "Here's the statement of my exercise, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
  – Shaun
  Jul 30 at 15:46
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I didn't ask anyone to 'solve it' but just some general ideas that could help me solving this. By your message I guess you didn't read the original post carefully. Regards
  – James Arten
  Jul 30 at 17:00
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @JamesArten. D'Oh. Of course.............
  – DanielWainfleet
  Jul 30 at 17:33
  

  
  
  
  

 | 
show 1 more comment

up vote
-2
down vote

favorite

Finally I decided to sign up to the forum which I've always found really helpful! I'm James and I'm a Math student (Master degree). Today I came across this exercise that I can't solve and I was wondering if any of you had some ideas about it. Basically I have:

$ mathbbQ = q_n_n in mathbbN$ where ($q_n ne q_m$ if $n ne m$) and let

$$ mu = sum_n in mathbbN frac1n^2 delta_q_n$$

Where $delta_x$ is Dirac delta centered in $x in mathbbR$.

Let $f : mathbbR rightarrow mathbbR$ such that
if $x gt 0$
$$f(x) = mu ((0,x)) $$
if $x le 0$
$$f(x) = mu ((x,0])$$

Summarize some $f$ properties (in particular, sign, continuity, right/left continuity, total variation, boundedness, monotony, values of $f(0^+)$, $f(0^-)$..) and some $mu$ properties (in particular (absolute continuity, discrete, singular.. respect to the Lebesgue measure). Thank you in advance!

real-analysis measure-theory dirac-delta

share|cite|improve this question

edited Jul 30 at 15:06

asked Jul 30 at 14:53

James Arten

186

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I don't understand your notation. For me $(0,x)$ is a point in $Bbb R^2$ and $mu ((0,x))$ to me is meaningless.
  – DanielWainfleet
  Jul 30 at 15:35
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  We're in $mathbbR$ so it's supposed to be the Measure of the interval $(0,x)$ for $x gt 0$. Is that right?
  – James Arten
  Jul 30 at 15:38
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  You'll find that simple "Here's the statement of my exercise, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
  – Shaun
  Jul 30 at 15:46
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I didn't ask anyone to 'solve it' but just some general ideas that could help me solving this. By your message I guess you didn't read the original post carefully. Regards
  – James Arten
  Jul 30 at 17:00
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @JamesArten. D'Oh. Of course.............
  – DanielWainfleet
  Jul 30 at 17:33
  

  
  
  
  

 | 
show 1 more comment

up vote
-2
down vote

favorite

up vote
-2
down vote

favorite

Finally I decided to sign up to the forum which I've always found really helpful! I'm James and I'm a Math student (Master degree). Today I came across this exercise that I can't solve and I was wondering if any of you had some ideas about it. Basically I have:

$ mathbbQ = q_n_n in mathbbN$ where ($q_n ne q_m$ if $n ne m$) and let

$$ mu = sum_n in mathbbN frac1n^2 delta_q_n$$

Where $delta_x$ is Dirac delta centered in $x in mathbbR$.

Let $f : mathbbR rightarrow mathbbR$ such that
if $x gt 0$
$$f(x) = mu ((0,x)) $$
if $x le 0$
$$f(x) = mu ((x,0])$$

Summarize some $f$ properties (in particular, sign, continuity, right/left continuity, total variation, boundedness, monotony, values of $f(0^+)$, $f(0^-)$..) and some $mu$ properties (in particular (absolute continuity, discrete, singular.. respect to the Lebesgue measure). Thank you in advance!

real-analysis measure-theory dirac-delta

share|cite|improve this question

edited Jul 30 at 15:06

asked Jul 30 at 14:53

James Arten

186

Finally I decided to sign up to the forum which I've always found really helpful! I'm James and I'm a Math student (Master degree). Today I came across this exercise that I can't solve and I was wondering if any of you had some ideas about it. Basically I have:

$ mathbbQ = q_n_n in mathbbN$ where ($q_n ne q_m$ if $n ne m$) and let

$$ mu = sum_n in mathbbN frac1n^2 delta_q_n$$

Where $delta_x$ is Dirac delta centered in $x in mathbbR$.

Let $f : mathbbR rightarrow mathbbR$ such that
if $x gt 0$
$$f(x) = mu ((0,x)) $$
if $x le 0$
$$f(x) = mu ((x,0])$$

Summarize some $f$ properties (in particular, sign, continuity, right/left continuity, total variation, boundedness, monotony, values of $f(0^+)$, $f(0^-)$..) and some $mu$ properties (in particular (absolute continuity, discrete, singular.. respect to the Lebesgue measure). Thank you in advance!

real-analysis measure-theory dirac-delta

share|cite|improve this question

edited Jul 30 at 15:06

asked Jul 30 at 14:53

James Arten

186

share|cite|improve this question

share|cite|improve this question

edited Jul 30 at 15:06

edited Jul 30 at 15:06

edited Jul 30 at 15:06

asked Jul 30 at 14:53

James Arten

186

asked Jul 30 at 14:53

James Arten

186

asked Jul 30 at 14:53

James Arten

186

186

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I don't understand your notation. For me $(0,x)$ is a point in $Bbb R^2$ and $mu ((0,x))$ to me is meaningless.
  – DanielWainfleet
  Jul 30 at 15:35
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  We're in $mathbbR$ so it's supposed to be the Measure of the interval $(0,x)$ for $x gt 0$. Is that right?
  – James Arten
  Jul 30 at 15:38
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  You'll find that simple "Here's the statement of my exercise, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
  – Shaun
  Jul 30 at 15:46
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I didn't ask anyone to 'solve it' but just some general ideas that could help me solving this. By your message I guess you didn't read the original post carefully. Regards
  – James Arten
  Jul 30 at 17:00
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @JamesArten. D'Oh. Of course.............
  – DanielWainfleet
  Jul 30 at 17:33
  

  
  
  
  

 | 
show 1 more comment

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I don't understand your notation. For me $(0,x)$ is a point in $Bbb R^2$ and $mu ((0,x))$ to me is meaningless.
  – DanielWainfleet
  Jul 30 at 15:35
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  We're in $mathbbR$ so it's supposed to be the Measure of the interval $(0,x)$ for $x gt 0$. Is that right?
  – James Arten
  Jul 30 at 15:38
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  You'll find that simple "Here's the statement of my exercise, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
  – Shaun
  Jul 30 at 15:46
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I didn't ask anyone to 'solve it' but just some general ideas that could help me solving this. By your message I guess you didn't read the original post carefully. Regards
  – James Arten
  Jul 30 at 17:00
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @JamesArten. D'Oh. Of course.............
  – DanielWainfleet
  Jul 30 at 17:33
  

  
  
  
  

I don't understand your notation. For me $(0,x)$ is a point in $Bbb R^2$ and $mu ((0,x))$ to me is meaningless.
– DanielWainfleet
Jul 30 at 15:35

I don't understand your notation. For me $(0,x)$ is a point in $Bbb R^2$ and $mu ((0,x))$ to me is meaningless.
– DanielWainfleet
Jul 30 at 15:35

1

1

We're in $mathbbR$ so it's supposed to be the Measure of the interval $(0,x)$ for $x gt 0$. Is that right?
– James Arten
Jul 30 at 15:38

We're in $mathbbR$ so it's supposed to be the Measure of the interval $(0,x)$ for $x gt 0$. Is that right?
– James Arten
Jul 30 at 15:38

1

1

You'll find that simple "Here's the statement of my exercise, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
– Shaun
Jul 30 at 15:46

You'll find that simple "Here's the statement of my exercise, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance.
– Shaun
Jul 30 at 15:46

I didn't ask anyone to 'solve it' but just some general ideas that could help me solving this. By your message I guess you didn't read the original post carefully. Regards
– James Arten
Jul 30 at 17:00

I didn't ask anyone to 'solve it' but just some general ideas that could help me solving this. By your message I guess you didn't read the original post carefully. Regards
– James Arten
Jul 30 at 17:00

@JamesArten. D'Oh. Of course.............
– DanielWainfleet
Jul 30 at 17:33

@JamesArten. D'Oh. Of course.............
– DanielWainfleet
Jul 30 at 17:33

 | 
show 1 more comment

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