The average of the items of $(0,10]$












1












$begingroup$


The question says it. Let $A = {,x ,|, 0 < x le 10,}$. What would be the average of all the items in this set? How do you prove it?
UPDATE
$x$ belongs to the set of real numbers.My thoughts:Is it possible to find the average of the items of $A = {x|0 <= x <= 10}$ where $x$ belongs to the set of real numbers?If it is, why couldn’t we use the same approach for my question?






means







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    What are your thoughts?
    $endgroup$
    – Christoph
    Dec 17 '18 at 20:53










  • $begingroup$
    To what set do the $x$'s belong? If it's $mathbb N$, then you will be able to add and average the items. But if it's $mathbb Q$ or $mathbb R$, you will have to change your question since these the interval for these 2 sets has an infinite number of elements.
    $endgroup$
    – Bernard Massé
    Dec 17 '18 at 20:59












  • $begingroup$
    $x$ belongs to the set of real numbers
    $endgroup$
    – tesryt ety
    Dec 17 '18 at 21:02










  • $begingroup$
    For a closed interval we have the average value function from calculus. But for open/non closed intervals this is interesting
    $endgroup$
    – T. Fo
    Dec 17 '18 at 21:05










  • $begingroup$
    Yes, it's possible. What kind of class do you need it for? Calculus? Probability theory?
    $endgroup$
    – Botond
    Dec 17 '18 at 21:11
















1












$begingroup$


The question says it. Let $A = {,x ,|, 0 < x le 10,}$. What would be the average of all the items in this set? How do you prove it?
UPDATE
$x$ belongs to the set of real numbers.My thoughts:Is it possible to find the average of the items of $A = {x|0 <= x <= 10}$ where $x$ belongs to the set of real numbers?If it is, why couldn’t we use the same approach for my question?






means







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    What are your thoughts?
    $endgroup$
    – Christoph
    Dec 17 '18 at 20:53










  • $begingroup$
    To what set do the $x$'s belong? If it's $mathbb N$, then you will be able to add and average the items. But if it's $mathbb Q$ or $mathbb R$, you will have to change your question since these the interval for these 2 sets has an infinite number of elements.
    $endgroup$
    – Bernard Massé
    Dec 17 '18 at 20:59












  • $begingroup$
    $x$ belongs to the set of real numbers
    $endgroup$
    – tesryt ety
    Dec 17 '18 at 21:02










  • $begingroup$
    For a closed interval we have the average value function from calculus. But for open/non closed intervals this is interesting
    $endgroup$
    – T. Fo
    Dec 17 '18 at 21:05










  • $begingroup$
    Yes, it's possible. What kind of class do you need it for? Calculus? Probability theory?
    $endgroup$
    – Botond
    Dec 17 '18 at 21:11














1












1








1





$begingroup$


The question says it. Let $A = {,x ,|, 0 < x le 10,}$. What would be the average of all the items in this set? How do you prove it?
UPDATE
$x$ belongs to the set of real numbers.My thoughts:Is it possible to find the average of the items of $A = {x|0 <= x <= 10}$ where $x$ belongs to the set of real numbers?If it is, why couldn’t we use the same approach for my question?






means







share|cite|improve this question











$endgroup$




The question says it. Let $A = {,x ,|, 0 < x le 10,}$. What would be the average of all the items in this set? How do you prove it?
UPDATE
$x$ belongs to the set of real numbers.My thoughts:Is it possible to find the average of the items of $A = {x|0 <= x <= 10}$ where $x$ belongs to the set of real numbers?If it is, why couldn’t we use the same approach for my question?






means




means






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 17 '18 at 21:02







tesryt ety

















asked Dec 17 '18 at 20:50









tesryt etytesryt ety

83




83








  • 1




    $begingroup$
    What are your thoughts?
    $endgroup$
    – Christoph
    Dec 17 '18 at 20:53










  • $begingroup$
    To what set do the $x$'s belong? If it's $mathbb N$, then you will be able to add and average the items. But if it's $mathbb Q$ or $mathbb R$, you will have to change your question since these the interval for these 2 sets has an infinite number of elements.
    $endgroup$
    – Bernard Massé
    Dec 17 '18 at 20:59












  • $begingroup$
    $x$ belongs to the set of real numbers
    $endgroup$
    – tesryt ety
    Dec 17 '18 at 21:02










  • $begingroup$
    For a closed interval we have the average value function from calculus. But for open/non closed intervals this is interesting
    $endgroup$
    – T. Fo
    Dec 17 '18 at 21:05










  • $begingroup$
    Yes, it's possible. What kind of class do you need it for? Calculus? Probability theory?
    $endgroup$
    – Botond
    Dec 17 '18 at 21:11














  • 1




    $begingroup$
    What are your thoughts?
    $endgroup$
    – Christoph
    Dec 17 '18 at 20:53










  • $begingroup$
    To what set do the $x$'s belong? If it's $mathbb N$, then you will be able to add and average the items. But if it's $mathbb Q$ or $mathbb R$, you will have to change your question since these the interval for these 2 sets has an infinite number of elements.
    $endgroup$
    – Bernard Massé
    Dec 17 '18 at 20:59












  • $begingroup$
    $x$ belongs to the set of real numbers
    $endgroup$
    – tesryt ety
    Dec 17 '18 at 21:02










  • $begingroup$
    For a closed interval we have the average value function from calculus. But for open/non closed intervals this is interesting
    $endgroup$
    – T. Fo
    Dec 17 '18 at 21:05










  • $begingroup$
    Yes, it's possible. What kind of class do you need it for? Calculus? Probability theory?
    $endgroup$
    – Botond
    Dec 17 '18 at 21:11








1




1




$begingroup$
What are your thoughts?
$endgroup$
– Christoph
Dec 17 '18 at 20:53




$begingroup$
What are your thoughts?
$endgroup$
– Christoph
Dec 17 '18 at 20:53












$begingroup$
To what set do the $x$'s belong? If it's $mathbb N$, then you will be able to add and average the items. But if it's $mathbb Q$ or $mathbb R$, you will have to change your question since these the interval for these 2 sets has an infinite number of elements.
$endgroup$
– Bernard Massé
Dec 17 '18 at 20:59






$begingroup$
To what set do the $x$'s belong? If it's $mathbb N$, then you will be able to add and average the items. But if it's $mathbb Q$ or $mathbb R$, you will have to change your question since these the interval for these 2 sets has an infinite number of elements.
$endgroup$
– Bernard Massé
Dec 17 '18 at 20:59














$begingroup$
$x$ belongs to the set of real numbers
$endgroup$
– tesryt ety
Dec 17 '18 at 21:02




$begingroup$
$x$ belongs to the set of real numbers
$endgroup$
– tesryt ety
Dec 17 '18 at 21:02












$begingroup$
For a closed interval we have the average value function from calculus. But for open/non closed intervals this is interesting
$endgroup$
– T. Fo
Dec 17 '18 at 21:05




$begingroup$
For a closed interval we have the average value function from calculus. But for open/non closed intervals this is interesting
$endgroup$
– T. Fo
Dec 17 '18 at 21:05












$begingroup$
Yes, it's possible. What kind of class do you need it for? Calculus? Probability theory?
$endgroup$
– Botond
Dec 17 '18 at 21:11




$begingroup$
Yes, it's possible. What kind of class do you need it for? Calculus? Probability theory?
$endgroup$
– Botond
Dec 17 '18 at 21:11










1 Answer
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$begingroup$

Considering $(0,10]$ or $[0,10]$ as subspaces of $mathbb R$ with the Lebesgue measure $mu$, the question of "average" is a measure theoretic (or probability theoretic) one. While average of a finite set $Xsubsetmathbb R$ may be defined as
$$
frac{1}{|X|} sum_{xin X} x,
$$
we can define the average of a set $Xsubsetmathbb R$ of finite measure $mu(X)<infty$ as the Lebesgue integral
$$
frac{1}{mu(X)} int_X x,mathrm dmu(x).
$$
When $X=(0,10]$ or $X=[0,10]$ you have $mu(X)=10$ and obtain the average
$$
frac{1}{10} int_0^{10} x,mathrm dx =5.
$$






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    $begingroup$

    Considering $(0,10]$ or $[0,10]$ as subspaces of $mathbb R$ with the Lebesgue measure $mu$, the question of "average" is a measure theoretic (or probability theoretic) one. While average of a finite set $Xsubsetmathbb R$ may be defined as
    $$
    frac{1}{|X|} sum_{xin X} x,
    $$
    we can define the average of a set $Xsubsetmathbb R$ of finite measure $mu(X)<infty$ as the Lebesgue integral
    $$
    frac{1}{mu(X)} int_X x,mathrm dmu(x).
    $$
    When $X=(0,10]$ or $X=[0,10]$ you have $mu(X)=10$ and obtain the average
    $$
    frac{1}{10} int_0^{10} x,mathrm dx =5.
    $$






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      Considering $(0,10]$ or $[0,10]$ as subspaces of $mathbb R$ with the Lebesgue measure $mu$, the question of "average" is a measure theoretic (or probability theoretic) one. While average of a finite set $Xsubsetmathbb R$ may be defined as
      $$
      frac{1}{|X|} sum_{xin X} x,
      $$
      we can define the average of a set $Xsubsetmathbb R$ of finite measure $mu(X)<infty$ as the Lebesgue integral
      $$
      frac{1}{mu(X)} int_X x,mathrm dmu(x).
      $$
      When $X=(0,10]$ or $X=[0,10]$ you have $mu(X)=10$ and obtain the average
      $$
      frac{1}{10} int_0^{10} x,mathrm dx =5.
      $$






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        Considering $(0,10]$ or $[0,10]$ as subspaces of $mathbb R$ with the Lebesgue measure $mu$, the question of "average" is a measure theoretic (or probability theoretic) one. While average of a finite set $Xsubsetmathbb R$ may be defined as
        $$
        frac{1}{|X|} sum_{xin X} x,
        $$
        we can define the average of a set $Xsubsetmathbb R$ of finite measure $mu(X)<infty$ as the Lebesgue integral
        $$
        frac{1}{mu(X)} int_X x,mathrm dmu(x).
        $$
        When $X=(0,10]$ or $X=[0,10]$ you have $mu(X)=10$ and obtain the average
        $$
        frac{1}{10} int_0^{10} x,mathrm dx =5.
        $$






        share|cite|improve this answer









        $endgroup$



        Considering $(0,10]$ or $[0,10]$ as subspaces of $mathbb R$ with the Lebesgue measure $mu$, the question of "average" is a measure theoretic (or probability theoretic) one. While average of a finite set $Xsubsetmathbb R$ may be defined as
        $$
        frac{1}{|X|} sum_{xin X} x,
        $$
        we can define the average of a set $Xsubsetmathbb R$ of finite measure $mu(X)<infty$ as the Lebesgue integral
        $$
        frac{1}{mu(X)} int_X x,mathrm dmu(x).
        $$
        When $X=(0,10]$ or $X=[0,10]$ you have $mu(X)=10$ and obtain the average
        $$
        frac{1}{10} int_0^{10} x,mathrm dx =5.
        $$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 17 '18 at 21:31









        ChristophChristoph

        12.5k1642




        12.5k1642






























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